package Math::Prime::Util;
use strict;
use warnings;
use Carp qw/croak confess carp/;

BEGIN {
  $Math::Prime::Util::AUTHORITY = 'cpan:DANAJ';
  $Math::Prime::Util::VERSION = '0.74';
}

# parent is cleaner, and in the Perl 5.10.1 / 5.12.0 core, but not earlier.
# use parent qw( Exporter );
use base qw( Exporter );
our @EXPORT_OK =
  qw( prime_get_config prime_set_config
      prime_precalc prime_memfree
      is_prime is_prob_prime is_provable_prime is_provable_prime_with_cert
      prime_certificate verify_prime
      is_pseudoprime is_euler_pseudoprime is_strong_pseudoprime
      is_euler_plumb_pseudoprime
      is_lucas_pseudoprime
      is_strong_lucas_pseudoprime
      is_extra_strong_lucas_pseudoprime
      is_almost_extra_strong_lucas_pseudoprime
      is_frobenius_pseudoprime
      is_frobenius_underwood_pseudoprime is_frobenius_khashin_pseudoprime
      is_perrin_pseudoprime is_catalan_pseudoprime
      is_aks_prime is_bpsw_prime is_ramanujan_prime is_mersenne_prime
      is_delicate_prime is_chen_prime
      is_odd is_even is_divisible is_congruent
      is_power is_prime_power is_perfect_power is_square
      is_square_free is_powerfree
      is_pillai is_polygonal is_congruent_number is_perfect_number
      is_semiprime is_almost_prime is_omega_prime
      is_primitive_root is_carmichael is_quasi_carmichael is_cyclic
      is_fundamental is_totient is_gaussian_prime is_sum_of_squares
      is_smooth is_rough is_powerful is_practical is_lucky is_happy
      sqrtint rootint logint lshiftint rshiftint rashiftint absint negint
      signint cmpint addint subint add1int sub1int mulint powint
      divint modint cdivint divrem fdivrem cdivrem tdivrem
      miller_rabin_random
      lucas_sequence
      lucasu lucasv lucasuv lucasumod lucasvmod lucasuvmod pisano_period
      primes twin_primes semi_primes almost_primes omega_primes ramanujan_primes
      sieve_prime_cluster sieve_range prime_powers lucky_numbers
      forprimes forcomposites foroddcomposites forsemiprimes foralmostprimes
      forpart forcomp forcomb forperm forderange formultiperm forsetproduct
      fordivisors forfactored forsquarefree forsquarefreeint
      lastfor
      numtoperm permtonum randperm shuffle vecsample
      prime_iterator prime_iterator_object
      next_prime prev_prime
      next_prime_power prev_prime_power
      next_perfect_power prev_perfect_power
      next_chen_prime
      prime_count prime_count_lower prime_count_upper prime_count_approx
      nth_prime nth_prime_lower nth_prime_upper nth_prime_approx
      inverse_li inverse_li_nv
      twin_prime_count twin_prime_count_approx
      nth_twin_prime nth_twin_prime_approx
      semiprime_count semiprime_count_approx
      nth_semiprime nth_semiprime_approx
      almost_prime_count almost_prime_count_approx
      almost_prime_count_lower almost_prime_count_upper
      nth_almost_prime nth_almost_prime_approx
      nth_almost_prime_lower nth_almost_prime_upper
      omega_prime_count nth_omega_prime
      ramanujan_prime_count ramanujan_prime_count_approx
      ramanujan_prime_count_lower ramanujan_prime_count_upper
      nth_ramanujan_prime nth_ramanujan_prime_approx
      nth_ramanujan_prime_lower nth_ramanujan_prime_upper
      powerful_count nth_powerful sumpowerful powerful_numbers
      prime_power_count prime_power_count_approx
      prime_power_count_lower prime_power_count_upper
      nth_prime_power nth_prime_power_approx
      nth_prime_power_lower nth_prime_power_upper
      perfect_power_count perfect_power_count_approx
      perfect_power_count_lower perfect_power_count_upper
      nth_perfect_power nth_perfect_power_approx
      nth_perfect_power_lower nth_perfect_power_upper
      nth_powerfree powerfree_count powerfree_sum squarefree_kernel
      powerfree_part powerfree_part_sum
      smooth_count rough_count powersum
      lucky_count lucky_count_approx lucky_count_lower lucky_count_upper
      nth_lucky nth_lucky_approx nth_lucky_lower nth_lucky_upper
      minimal_goldbach_pair goldbach_pairs goldbach_pair_count
      sum_primes print_primes
      random_prime random_ndigit_prime
      random_nbit_prime random_safe_prime random_strong_prime
      random_proven_prime random_proven_prime_with_cert
      random_maurer_prime random_maurer_prime_with_cert
      random_shawe_taylor_prime random_shawe_taylor_prime_with_cert
      random_semiprime random_unrestricted_semiprime
      random_factored_integer
      primorial pn_primorial consecutive_integer_lcm gcdext chinese chinese2
      gcd lcm factor factor_exp divisors valuation hammingweight
      frobenius_number
      todigits fromdigits todigitstring sumdigits
      tozeckendorf fromzeckendorf
      sqrtmod allsqrtmod rootmod allrootmod cornacchia
      negmod invmod addmod submod mulmod divmod powmod muladdmod mulsubmod
      vecsum vecmin vecmax vecprod vecreduce vecextract vecequal vecuniq
      vecany vecall vecnotall vecnone vecfirst vecfirstidx vecmex vecpmex
      vecsort vecsorti vecfreq vecsingleton vecslide
      setbinop sumset toset
      setunion setintersect setminus setdelta
      setcontains setcontainsany setinsert setremove setinvert
      is_sidon_set is_sumfree_set
      set_is_disjoint set_is_equal set_is_proper_intersection
      set_is_subset set_is_proper_subset set_is_superset set_is_proper_superset
      moebius mertens liouville sumliouville prime_omega prime_bigomega
      euler_phi jordan_totient exp_mangoldt sumtotient
      partitions bernfrac bernreal harmfrac harmreal
      chebyshev_theta chebyshev_psi
      divisor_sum carmichael_lambda hclassno inverse_totient
      kronecker is_qr qnr
      ramanujan_tau ramanujan_sum
      stirling fubini znorder znprimroot znlog legendre_phi
      factorial factorialmod subfactorial binomial binomialmod
      falling_factorial rising_factorial
      contfrac from_contfrac
      next_calkin_wilf next_stern_brocot
      calkin_wilf_n stern_brocot_n
      nth_calkin_wilf nth_stern_brocot
      nth_stern_diatomic
      farey next_farey farey_rank
      ExponentialIntegral LogarithmicIntegral RiemannZeta RiemannR LambertW Pi
      irand irand64 drand urandomb urandomm csrand random_bytes entropy_bytes
  );
our %EXPORT_TAGS = (all  => [ @EXPORT_OK ],
                    rand => [qw/srand rand irand irand64/],
                   );

# These are only exported if specifically asked for
push @EXPORT_OK, (qw/trial_factor fermat_factor holf_factor lehman_factor squfof_factor prho_factor pbrent_factor pminus1_factor pplus1_factor cheb_factor ecm_factor rand srand/);

my %_Config;
my %_GMPfunc;  # Available MPU::GMP functions

# Similar to how boolean handles its option
sub import {
    if ($] < 5.020) {   # Prevent "used only once" warnings
      my $pkg = caller;
      no strict 'refs';  ## no critic(strict)
      ${"${pkg}::a"} = ${"${pkg}::a"};
      ${"${pkg}::b"} = ${"${pkg}::b"};
    }
    foreach my $opt (qw/nobigint secure/) {
      my @options = grep $_ ne "-$opt", @_;
      $_Config{$opt} = 1 if @options != @_;
      @_ = @options;
    }
    _XS_set_secure() if $_Config{'xs'} && $_Config{'secure'};
    goto &Exporter::import;
}

#############################################################################


BEGIN {

  # Separate lines to keep compatible with default from 5.6.2.
  # We could alternately use Config's $Config{uvsize} for MAXBITS
  use constant OLD_PERL_VERSION=> $] < 5.008;
  use constant MPU_MAXBITS     => (~0 == 4294967295) ? 32 : 64;
  use constant MPU_32BIT       => MPU_MAXBITS == 32;
  use constant MPU_MAXPARAM    => MPU_32BIT ? 4294967295 : 18446744073709551615;
  use constant MPU_MAXDIGITS   => MPU_32BIT ?         10 : 20;
  use constant MPU_MAXPRIME    => MPU_32BIT ? 4294967291 : 18446744073709551557;
  use constant MPU_MAXPRIMEIDX => MPU_32BIT ?  203280221 :   425656284035217743;
  use constant UVPACKLET       => MPU_32BIT ?        'L' : 'Q';
  use constant INTMAX          => (!OLD_PERL_VERSION || MPU_32BIT) ? ~0 : 562949953421312;
  use constant INTMIN          => -(INTMAX >> 1) - 1;

  eval {
    return 0 if defined $ENV{MPU_NO_XS} && $ENV{MPU_NO_XS} == 1;
    require XSLoader;
    XSLoader::load(__PACKAGE__, $Math::Prime::Util::VERSION);
    prime_precalc(0);
    $_Config{'maxbits'} = _XS_prime_maxbits();
    $_Config{'xs'} = 1;
    1;
  } or do {
    carp "Using Pure Perl implementation: $@"
      unless defined $ENV{MPU_NO_XS} && $ENV{MPU_NO_XS} == 1;

    $_Config{'xs'} = 0;
    $_Config{'maxbits'} = MPU_MAXBITS;

    # Load PP front end code
    require Math::Prime::Util::PPFE;

    # Init rand
    Math::Prime::Util::csrand();
  };

  $_Config{'secure'} = 0;
  $_Config{'nobigint'} = 0;
  $_Config{'gmp'} = 0;
  # See if they have the GMP module and haven't requested it not to be used.
  if (!defined $ENV{MPU_NO_GMP} || $ENV{MPU_NO_GMP} != 1) {
    if (eval { require Math::Prime::Util::GMP;
               Math::Prime::Util::GMP->import();
               1; }) {
      $_Config{'gmp'} = int(100 * $Math::Prime::Util::GMP::VERSION + 1e-6);
    }
    for my $e (@Math::Prime::Util::GMP::EXPORT_OK) {
      $Math::Prime::Util::_GMPfunc{"$e"} = $_Config{'gmp'};
    }
    # If we have GMP, it is not seeded properly but we are, seed with our data.
    if (   $_Config{'gmp'} >= 42
        && !Math::Prime::Util::GMP::is_csprng_well_seeded()
        && Math::Prime::Util::_is_csprng_well_seeded()) {
      Math::Prime::Util::GMP::seed_csprng(256, random_bytes(256));
    }
  }

  # Alias PP and GMP if requested.  Very convenient but a big hammer.
  if (defined $ENV{MPU_DEVNAMES} && $ENV{MPU_DEVNAMES} == 1) {
    no strict 'refs';  ## no critic(strict)
    *MPU:: = \*Math::Prime::Util::;
    *PP:: = \*Math::Prime::Util::PP:: if eval { require Math::Prime::Util::PP; Math::Prime::Util::PP->import(); 1; };
    *GMP:: = \*Math::Prime::Util::GMP:: if $_Config{'gmp'};
  }
}

croak "Perl and XS don't agree on bit size"
      if $_Config{'xs'} && MPU_MAXBITS != _XS_prime_maxbits();

$_Config{'maxparam'}    = MPU_MAXPARAM;
$_Config{'maxdigits'}   = MPU_MAXDIGITS;
$_Config{'maxprime'}    = MPU_MAXPRIME;
$_Config{'maxprimeidx'} = MPU_MAXPRIMEIDX;
$_Config{'assume_rh'}   = 0;
$_Config{'verbose'}     = 0;
$_Config{'bigintclass'} = undef;

# used for code like:
#    return _XS_foo($n)  if $n <= $_XS_MAXVAL
# which builds into one scalar whether XS is available and if we can call it.
my $_XS_MAXVAL = $_Config{'xs'}  ?  MPU_MAXPARAM  :  -1;
my $_HAVE_GMP = $_Config{'gmp'};
_XS_set_callgmp($_HAVE_GMP) if $_Config{'xs'};

our $_BIGINT = $_Config{'bigintclass'};

# Infinity in Perl is rather O/S specific.
our $_Infinity = 0+'inf';
$_Infinity = 20**20**20 if 65535 > $_Infinity;   # E.g. Windows
our $_Neg_Infinity = -$_Infinity;

sub prime_get_config {
  my %config = %_Config;

  $config{'precalc_to'} = ($_Config{'xs'})
                        ? _get_prime_cache_size()
                        : Math::Prime::Util::PP::_get_prime_cache_size();

  return \%config;
}

# Note: You can cause yourself pain if you turn on xs or gmp when they're not
# loaded.  Your calls will probably die horribly.
sub prime_set_config {
  my %params = (@_);  # no defaults
  foreach my $param (keys %params) {
    my $value = $params{$param};
    $param = lc $param;
    # dispatch table should go here.
    if      ($param eq 'xs') {
      $_Config{'xs'} = ($value) ? 1 : 0;
      $_XS_MAXVAL = $_Config{'xs'}  ?  MPU_MAXPARAM  :  -1;
    } elsif ($param eq 'gmp') {
      $_HAVE_GMP = ($value) ? int(100*$Math::Prime::Util::GMP::VERSION) : 0;
      $_Config{'gmp'} = $_HAVE_GMP;
      $Math::Prime::Util::_GMPfunc{$_} = $_HAVE_GMP
        for keys %Math::Prime::Util::_GMPfunc;
      _XS_set_callgmp($_HAVE_GMP) if $_Config{'xs'};
    } elsif ($param eq 'nobigint') {
      $_Config{'nobigint'} = ($value) ? 1 : 0;
    } elsif ($param eq 'bigint' || $param eq 'trybigint') {
      my $class = _load_bigint_class($value);
      if (defined $class) {
        $_BIGINT = $_Config{'bigintclass'} = $class;
      } else {
        carp "ntheory could not load bigint class from '$value'"
          unless $param =~ /try/;
      }
    } elsif ($param eq 'secure') {
      croak "Cannot disable secure once set" if !$value && $_Config{'secure'};
      if ($value) {
        $_Config{'secure'} = 1;
        _XS_set_secure() if $_Config{'xs'};
      }
    } elsif ($param eq 'irand') {
      carp "ntheory irand option is deprecated";
    } elsif ($param eq 'use_primeinc') {
      carp "ntheory use_primeinc option is deprecated";
    } elsif ($param =~ /^(assume[_ ]?)?[ge]?rh$/ || $param =~ /riemann\s*h/) {
      $_Config{'assume_rh'} = ($value) ? 1 : 0;
    } elsif ($param eq 'verbose') {
      if    ($value =~ /^\d+$/) { }
      elsif ($value =~ /^[ty]/i) { $value = 1; }
      elsif ($value =~ /^[fn]/i) { $value = 0; }
      else { croak("Invalid setting for verbose.  0, 1, 2, etc."); }
      $_Config{'verbose'} = $value;
      _XS_set_verbose($value) if $_Config{'xs'};
      Math::Prime::Util::GMP::_GMP_set_verbose($value) if $_Config{'gmp'};
    } else {
      croak "Unknown or invalid configuration setting: $param\n";
    }
  }
  1;
}

# Input:  object, or comma separated list of class names
# Output: class name or undef
sub _load_bigint_class {
  my($val) = @_;

  my $class = undef;
  if (ref($val)) {      # We are given an object, e.g. a Math::GMPz number
    $class = ref($val);
  } else {              # Comma separated list of class names
    for my $name (split /,/, $val) {
      $name =~ s/^\s+|\s+$//g;
      (my $cfname="$name.pm")=~s|::|/|g; # Foo::Bar::Baz => Foo/Bar/Baz.pm
      if ($INC{$cfname} || eval { require $cfname; $name->import(); 1; }) {
        $class = $name;
        last;
      }
    }
  }
  if ($class) {         # Check we can make a number with it
    $class = undef unless eval { $class->new(1) == 1 };
  }
  return $class;
}

# This is for loading the default bigint class the very first time.
sub _load_bigint {
  return $_BIGINT if defined $_BIGINT;

  # TODO: turn this on for next release
  #prime_set_config( trybigint => 'Math::GMPz,Math::GMP' );
  #return $_BIGINT if defined $_BIGINT;

  do { require Math::BigInt;  Math::BigInt->import(try=>"GMP,GMPz,LTM,Pari"); } unless defined $Math::BigInt::VERSION;
  $_BIGINT = $_Config{'bigintclass'} = 'Math::BigInt';
}

sub _bigint_to_int {
  return (OLD_PERL_VERSION && $_[0] >= 0) ? unpack(UVPACKLET,pack(UVPACKLET,"$_[0]"))
                            : int("$_[0]");
}
sub _to_bigint {
  return undef unless defined($_[0]);
  _load_bigint() unless defined $_BIGINT;
  # We don't do any validation other than that the class is happy.
  my $n;
  if (ref($_[0]) eq $_BIGINT) {
    $n = $_[0];
  } elsif (ref($_[0]) eq 'Math::BigFloat' && !$_[0]->is_int()) {
    $n = Math::BigInt->bnan;
  } elsif ($_BIGINT eq 'Math::Pari' && $_[0] =~ /^0[bx]/) {
    # Pari added support for this in 2.8, so not in Math::Pari
    do { require Math::BigInt;  Math::BigInt->import(try=>"GMP,GMPz,LTM,Pari"); } unless defined $Math::BigInt::VERSION;
    $n = Math::BigInt->new("$_[0]");
    $n = $_BIGINT->new("$n");
  } else {
    $n = $_BIGINT->new("$_[0]");
  }
  croak "Parameter '$_[0]' must be an integer" unless $_BIGINT ne 'Math::BigInt' || $n->is_int();
  $n;
}
sub _to_bigint_nonneg {
  return undef unless defined($_[0]);
  _load_bigint() unless defined $_BIGINT;
  my $n;
  if (ref($_[0]) eq $_BIGINT) {
    $n = $_[0];
  } elsif (ref($_[0]) eq 'Math::BigFloat' && !$_[0]->is_int()) {
    $n = Math::BigInt->bnan;
  } else {
    $n = $_BIGINT->new("$_[0]");
  }
  croak "Parameter '$_[0]' must be a non-negative integer" unless ($_BIGINT ne 'Math::BigInt' || $n->is_int()) && $n >= 0;
  $n;
}
sub _to_bigint_abs {
  return undef unless defined($_[0]);
  my $n = _to_bigint($_[0]);
  return ($n < 0) ? -$n : $n;
}
sub _to_bigint_if_needed {
  return $_[0] if !defined $_[0] || ref($_[0]);
  if ($_[0] >= INTMAX || $_[0] <= INTMIN) {    # Probably a bigint
    my $n = _to_bigint($_[0]);
    return $n if $n > INTMAX || $n < INTMIN;   # Definitely a bigint
  }
  $_[0];
}
sub _to_gmpz {
  do { require Math::GMPz; } unless defined $Math::GMPz::VERSION;
  return (ref($_[0]) eq 'Math::GMPz') ? $_[0] : Math::GMPz->new($_[0]);
}
sub _to_gmp {
  do { require Math::GMP; } unless defined $Math::GMP::VERSION;
  return (ref($_[0]) eq 'Math::GMP') ? $_[0] : Math::GMP->new($_[0]);
}
sub _reftyped {
  return undef unless defined $_[1];
  my $ref0 = ref($_[0]);
  if (OLD_PERL_VERSION) {
    # Perl 5.6 truncates arguments to doubles if you look at them funny
    return "$_[1]" if "$_[1]" <= INTMAX && "$_[1]" >= INTMIN;
  } elsif ($_[1] >= 0) {
    return $_[1] if $_[1] < ~0;
  } else {
    return $_[1] if $_[1] > -(~0>>1);
  }
  my $bign;
  if ($ref0) {
    $bign = $ref0->new("$_[1]");
  } else {
    _load_bigint() unless defined $_BIGINT;
    $bign = $_BIGINT->new("$_[1]");
  }
  return $bign if $bign > INTMAX || $bign < INTMIN;
  0+"$_[1]";
}

sub _maybe_bigint_allargs {
  _load_bigint() unless defined $_BIGINT;
  for my $i (0..$#_) {
    next if !defined $_[$i] || ref($_[$i]);
    next if $_[$i] < INTMAX && $_[$i] > INTMIN;
    my $n = $_BIGINT->new("$_[$i]");
    $_[$i] = $n if $n > INTMAX || $n < INTMIN;
  }
  @_;
}

#############################################################################

# These are called by the XS code to keep the GMP CSPRNG in sync with us.

sub _srand_p {
  my($seedval) = @_;
  return unless $_Config{'gmp'} >= 42;
  $seedval = unpack("L",entropy_bytes(4)) unless defined $seedval;
  Math::Prime::Util::GMP::seed_csprng(4, pack("L",$seedval));
  $seedval;
}

sub _csrand_p {
  my($str) = @_;
  return unless $_Config{'gmp'} >= 42;
  $str = entropy_bytes(256) unless defined $str;
  Math::Prime::Util::GMP::seed_csprng(length($str), $str);
}

#############################################################################


#############################################################################
# Random primes.  These are front end functions that do input validation,
# load the RandomPrimes module, and call its function.

sub random_maurer_prime_with_cert {
  my($bits) = @_;
  _validate_integer_nonneg($bits);
  croak "random_maurer_prime bits must be >= 2" unless $bits >= 2;

  if ($Math::Prime::Util::_GMPfunc{"random_maurer_prime_with_cert"}) {
    my($n,$cert) = Math::Prime::Util::GMP::random_maurer_prime_with_cert($bits);
    return (Math::Prime::Util::_reftyped($_[0],$n), $cert);
  }

  require Math::Prime::Util::RandomPrimes;
  return Math::Prime::Util::RandomPrimes::random_maurer_prime_with_cert($bits);
}

sub random_shawe_taylor_prime_with_cert {
  my($bits) = @_;
  _validate_integer_nonneg($bits);
  croak "random_shawe_taylor_prime bits must be >= 2" unless $bits >= 2;

  if ($Math::Prime::Util::_GMPfunc{"random_shawe_taylor_prime_with_cert"}) {
    my($n,$cert) =Math::Prime::Util::GMP::random_shawe_taylor_prime_with_cert($bits);
    return (Math::Prime::Util::_reftyped($_[0],$n), $cert);
  }

  require Math::Prime::Util::RandomPrimes;
  return Math::Prime::Util::RandomPrimes::random_shawe_taylor_prime_with_cert($bits);
}

sub random_proven_prime_with_cert {
  my($bits) = @_;
  _validate_integer_nonneg($bits);
  croak "random_proven_prime bits must be >= 2" unless $bits >= 2;

  # Go to Maurer with GMP
  if ($Math::Prime::Util::_GMPfunc{"random_maurer_prime_with_cert"}) {
    my($n,$cert) = Math::Prime::Util::GMP::random_maurer_prime_with_cert($bits);
    return (Math::Prime::Util::_reftyped($_[0],$n), $cert);
  }

  require Math::Prime::Util::RandomPrimes;
  return Math::Prime::Util::RandomPrimes::random_proven_prime_with_cert($bits);
}

#############################################################################

sub formultiperm (&$) {    ## no critic qw(ProhibitSubroutinePrototypes)
  my($sub, $iref) = @_;
  croak("formultiperm first argument must be an array reference") unless ref($iref) eq 'ARRAY';

  my($sum, %h, @n) = (0);
  $h{$_}++ for @$iref;
  @n = map { [$_, $h{$_}] } sort(keys(%h));
  $sum += $_->[1] for @n;

  require Math::Prime::Util::PP;
  my $oldforexit = Math::Prime::Util::_start_for_loop();
  Math::Prime::Util::PP::_multiset_permutations( $sub, [], \@n, $sum );
  Math::Prime::Util::_end_for_loop($oldforexit);
}

#############################################################################
# Iterators

sub prime_iterator {
  my($start) = @_;
  $start = 0 unless defined $start;
  _validate_integer_nonneg($start);
  my $p = ($start > 0) ? $start-1 : 0;
  # This works fine:
  #   return sub { $p = next_prime($p); return $p; };
  # but we can optimize a little
  if (!ref($p) && $p <= $_XS_MAXVAL) {
    # This is simple and low memory, but slower than segments:
    #     return sub { $p = next_prime($p); return $p; };
    my $pr = [];
    return sub {
      if (scalar(@$pr) == 0) {
        # Once we're into bigints, just use next_prime
        return $p=next_prime($p) if $p >= MPU_MAXPRIME;
        # Get about 10k primes
        my $segment = ($p <= 1e4) ? 10_000 : int(10000*log($p)+1);
        $segment = ~0-$p if $p+$segment > ~0 && $p+1 < ~0;
        $pr = primes($p+1, $p+$segment);
      }
      return $p = shift(@$pr);
    };
  } elsif ($_HAVE_GMP) {
    return sub { $p = addint(0,Math::Prime::Util::GMP::next_prime($p)); return $p;};
  } else {
    require Math::Prime::Util::PP;
    return sub { $p = Math::Prime::Util::PP::next_prime($p); return $p; }
  }
}

sub prime_iterator_object {
  my($start) = @_;
  require Math::Prime::Util::PrimeIterator;
  return Math::Prime::Util::PrimeIterator->new($start);
}

#############################################################################
# Front ends to functions.
#
# These will do input validation, then call the appropriate internal function
# based on the input (XS, GMP, PP).
#############################################################################

#############################################################################

# Return just the cert portion.
sub prime_certificate {
  my($n) = @_;
  my ($is_prime, $cert) = is_provable_prime_with_cert($n);
  return $cert;
}


sub is_provable_prime_with_cert {
  my($n) = @_;
  _validate_integer($n);
  return 0 if $n < 2;
  my $header = "[MPU - Primality Certificate]\nVersion 1.0\n\nProof for:\nN $n\n\n";

  if ($n <= $_XS_MAXVAL) {
    my $isp = is_prime($n);
    return ($isp, '') unless $isp == 2;
    return (2, $header . "Type Small\nN $n\n");
  }

  if ($_HAVE_GMP && defined &Math::Prime::Util::GMP::is_provable_prime_with_cert) {
    my ($isp, $cert) = Math::Prime::Util::GMP::is_provable_prime_with_cert($n);
    # New version that returns string format.
    #return ($isp, $cert) if ref($cert) ne 'ARRAY';
    if (ref($cert) ne 'ARRAY') {
      # Fix silly 0.13 mistake (TODO: deprecate this)
      $cert =~ s/^Type Small\n(\d+)/Type Small\nN $1/smg;
      return ($isp, $cert);
    }
    # Old version.  Convert.
    require Math::Prime::Util::PrimalityProving;
    return ($isp, Math::Prime::Util::PrimalityProving::convert_array_cert_to_string($cert));
  }

  {
    my $isp = is_prob_prime($n);
    return ($isp, '') if $isp == 0;
    return (2, $header . "Type Small\nN $n\n") if $isp == 2;
  }

  # Choice of methods for proof:
  #   ECPP         needs a fair bit of programming work
  #   APRCL        needs a lot of programming work
  #   BLS75 combo  Corollary 11 of BLS75.  Trial factor n-1 and n+1 to B, find
  #                factors F1 of n-1 and F2 of n+1.  Quit when:
  #                B > (N/(F1*F1*(F2/2)))^1/3 or B > (N/((F1/2)*F2*F2))^1/3
  #   BLS75 n+1    Requires factoring n+1 to (n/2)^1/3 (theorem 19)
  #   BLS75 n-1    Requires factoring n-1 to (n/2)^1/3 (theorem 5 or 7)
  #   Pocklington  Requires factoring n-1 to n^1/2 (BLS75 theorem 4)
  #   Lucas        Easy, requires factoring of n-1 (BLS75 theorem 1)
  #   AKS          horribly slow
  # See http://primes.utm.edu/prove/merged.html or other sources.

  require Math::Prime::Util::PrimalityProving;
  #my ($isp, $pref) = Math::Prime::Util::PrimalityProving::primality_proof_lucas($n);
  my ($isp, $pref) = Math::Prime::Util::PrimalityProving::primality_proof_bls75($n);
  carp "proved $n is not prime\n" if !$isp;
  return ($isp, $pref);
}


sub verify_prime {
  require Math::Prime::Util::PrimalityProving;
  return Math::Prime::Util::PrimalityProving::verify_cert(@_);
}

#############################################################################

sub RiemannZeta {
  my($n) = @_;
  croak("Invalid input to RiemannZeta:  x must be > 0") if $n <= 0;

  return $n-$n if $n > 10_000_000;   # Over 3M leading zeros

  return _XS_RiemannZeta($n) if !ref($n) && $_Config{'xs'};

  require Math::Prime::Util::PP;
  return Math::Prime::Util::PP::RiemannZeta($n);
}

sub RiemannR {
  my($n) = @_;
  croak("Invalid input to RiemannR:  x must be > 0") if $n <= 0;

  return _XS_RiemannR($n) if !ref($n) && $_Config{'xs'};

  require Math::Prime::Util::PP;
  return Math::Prime::Util::PP::RiemannR($n);
}

sub ExponentialIntegral {
  my($n) = @_;
  return $_Neg_Infinity if $n == 0;
  return 0              if $n == $_Neg_Infinity;
  return $_Infinity     if $n == $_Infinity;

  return _XS_ExponentialIntegral($n) if !ref($n) && $_Config{'xs'};

  require Math::Prime::Util::PP;
  return Math::Prime::Util::PP::ExponentialIntegral($n);
}

sub LogarithmicIntegral {
  my($n) = @_;
  return 0              if $n == 0;
  return $_Neg_Infinity if $n == 1;
  return $_Infinity     if $n == $_Infinity;

  croak("Invalid input to LogarithmicIntegral:  x must be >= 0") if $n <= 0;

  if (!ref($n) && $_Config{'xs'}) {
    return 1.045163780117492784844588889194613136522615578151 if $n == 2;
    return _XS_LogarithmicIntegral($n);
  }

  require Math::Prime::Util::PP;
  return Math::Prime::Util::PP::LogarithmicIntegral(@_);
}

sub LambertW {
  my($x) = @_;

  return _XS_LambertW($x) if !ref($x) && $_Config{'xs'};

  require Math::Prime::Util::PP;
  return Math::Prime::Util::PP::LambertW($x);
}

sub bernreal {
  my($n, $precision) = @_;
  do { require Math::BigFloat; Math::BigFloat->import(); } unless defined $Math::BigFloat::VERSION;

  if ($Math::Prime::Util::_GMPfunc{"bernreal"}) {
    return Math::BigFloat->new(Math::Prime::Util::GMP::bernreal($n)) if !defined $precision;
    return Math::BigFloat->new(Math::Prime::Util::GMP::bernreal($n,$precision),$precision);
  }

  my($num,$den) = map { _to_bigint($_) } bernfrac($n);
  return Math::BigFloat->bzero if $num == 0;
  scalar Math::BigFloat->new($num)->bdiv($den, $precision);
}

sub harmreal {
  my($n, $precision) = @_;
  _validate_integer_nonneg($n);
  do { require Math::BigFloat; Math::BigFloat->import(); } unless defined $Math::BigFloat::VERSION;
  return Math::BigFloat->bzero if $n <= 0;

  if ($Math::Prime::Util::_GMPfunc{"harmreal"}) {
    return Math::BigFloat->new(Math::Prime::Util::GMP::harmreal($n)) if !defined $precision;
    return Math::BigFloat->new(Math::Prime::Util::GMP::harmreal($n,$precision),$precision);
  }

  # If low enough precision, use native floating point.  Fast.
  if (defined $precision && $precision <= 13) {
    return Math::BigFloat->new(
      ($n < 80) ? do { my $h = 0; $h += 1/$_ for 1..$n; $h; }
                : log($n) + 0.57721566490153286060651209 + 1/(2*$n) - 1/(12*$n*$n) + 1/(120*$n*$n*$n*$n)
      ,$precision
    );
  }

  if ($Math::Prime::Util::_GMPfunc{"harmfrac"}) {
    my($num,$den) = map { _to_bigint($_) } Math::Prime::Util::GMP::harmfrac($n);
    return scalar Math::BigFloat->new($num)->bdiv($den, $precision);
  }

  require Math::Prime::Util::PP;
  Math::Prime::Util::PP::harmreal($n, $precision);
}



#############################################################################

1;

__END__


# ABSTRACT: Utilities related to prime numbers, including fast generators / sievers

=pod

=encoding utf8

=for stopwords Möbius Deléglise Bézout uniqued k-tuples von SoE primesieve primegen libtommath pari yafu fonction qui compte le nombre nombres voor PhD superset sqrt(N) gcd(A^M k-th (10001st untruncated OpenPFGW gmpy2 Über Primzahl-Zählfunktion n-te und verallgemeinerte multiset compositeness GHz significand TestU01 subfactorial s-gonal XSLoader setwise

=for test_synopsis use v5.14;  my($k,$x);


=head1 NAME

Math::Prime::Util - Utilities related to prime numbers, including fast sieves and factoring


=head1 VERSION

Version 0.74


=head1 SYNOPSIS

  # Nothing is exported by default.  List the functions, or use :all.
  use Math::Prime::Util ':all';  # import all functions

  # The ':rand' tag replaces srand and rand (not done by default)
  use Math::Prime::Util ':rand';  # import srand, rand, irand, irand64


  # Get a big array reference of many primes
  my $aref = primes( 100_000_000 );

  # All the primes between 5k and 10k inclusive
  $aref = primes( 5_000, 10_000 );

  # If you want them in an array instead
  my @primes = @{primes( 500 )};

  # You can do something for every prime in a range.  Twin primes to 10k:
  forprimes { say if is_prime($_+2) } 10000;
  # Or for the composites in a range
  forcomposites { say if is_strong_pseudoprime($_,2) } 10000, 10**6;

  # For non-bigints, is_prime and is_prob_prime will always be 0 or 2.
  # They return 0 (composite), 2 (prime), or 1 (probably prime)
  my $n = 1000003;  # for example
  say "$n is prime"  if is_prime($n);
  say "$n is ", (qw(composite maybe_prime? prime))[is_prob_prime($n)];

  # Strong pseudoprime test with multiple bases, using Miller-Rabin
  say "$n is a prime or 2/7/61-psp" if is_strong_pseudoprime($n, 2, 7, 61);

  # Standard and strong Lucas-Selfridge, and extra strong Lucas tests
  say "$n is a prime or lpsp"   if is_lucas_pseudoprime($n);
  say "$n is a prime or slpsp"  if is_strong_lucas_pseudoprime($n);
  say "$n is a prime or eslpsp" if is_extra_strong_lucas_pseudoprime($n);

  # step to the next prime (returns 0 if not using bigints and we'd overflow)
  $n = next_prime($n);

  # step back (returns undef if given input 2 or less)
  $n = prev_prime($n);


  # Return Pi(n) -- the number of primes E<lt>= n.
  my $primepi = prime_count( 1_000_000 );
  $primepi = prime_count( 10**14, 10**14+1000 );  # also does ranges

  # Quickly return an approximation to Pi(n)
  my $approx_number_of_primes = prime_count_approx( 10**17 );

  # Lower and upper bounds.  lower <= Pi(n) <= upper for all n
  die unless prime_count_lower($n) <= prime_count($n);
  die unless prime_count_upper($n) >= prime_count($n);


  # Return p_n, the nth prime
  say "The ten thousandth prime is ", nth_prime(10_000);

  # Return a quick approximation to the nth prime
  say "The one trillionth prime is ~ ", nth_prime_approx(10**12);

  # Lower and upper bounds.   lower <= nth_prime(n) <= upper for all n
  die unless nth_prime_lower($n) <= nth_prime($n);
  die unless nth_prime_upper($n) >= nth_prime($n);


  # Get the prime factors of a number
  my @prime_factors = factor( $n );

  # Return ([p1,e1],[p2,e2], ...) for $n = p1^e1 * p2^e2 * ...
  my @pe = factor_exp( $n );

  # Get all divisors including 1 and n
  my @divisors = divisors( $n );
  # Or just apply a block for each one
  my $sum = 0; fordivisors  { $sum += $_ + $_*$_ }  $n;

  # Euler phi (Euler's totient) on a large number
  use bigint;  say euler_phi( 801294088771394680000412 );
  say jordan_totient(5, 1234);  # Jordan's totient

  # Moebius function used to calculate Mertens
  $sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
  # Mertens function directly (more efficient for large values)
  say mertens(10_000_000);
  # Exponential of Mangoldt function
  say "lamba(49) = ", log(exp_mangoldt(49));
  # Some more number theoretical functions
  say liouville(4292384);
  say chebyshev_psi(234984);
  say chebyshev_theta(92384234);
  say partitions(1000);
  # Show all prime partitions of 25
  forpart { say "@_" unless scalar grep { !is_prime($_) } @_ } 25;
  # List all 3-way combinations of an array
  my @cdata = qw/apple bread curry donut eagle/;
  forcomb { say "@cdata[@_]" } @cdata, 3;
  # or all permutations
  forperm { say "@cdata[@_]" } @cdata;

  # divisor sum
  my $sigma  = divisor_sum( $n );       # sum of divisors
  my $sigma0 = divisor_sum( $n, 0 );    # count of divisors
  my $sigmak = divisor_sum( $n, $k );
  my $sigmaf = divisor_sum( $n, sub { log($_[0]) } ); # arbitrary func

  # primorial n#, primorial p(n)#, and lcm
  say "The product of primes below 47 is ",     primorial(47);
  say "The product of the first 47 primes is ", pn_primorial(47);
  say "lcm(1..1000) is ", consecutive_integer_lcm(1000);

  # Ei, li, and Riemann R functions
  my $ei   = ExponentialIntegral($x);   # $x a real: $x != 0
  my $li   = LogarithmicIntegral($x);   # $x a real: $x >= 0
  my $R    = RiemannR($x);              # $x a real: $x > 0
  my $Zeta = RiemannZeta($x);           # $x a real: $x >= 0


  # Precalculate a sieve, possibly speeding up later work.
  prime_precalc( 1_000_000_000 );

  # Free any memory used by the module.
  prime_memfree;

  # Alternate way to free.  When this leaves scope, memory is freed.
  use Math::Prime::Util::MemFree;
  my $mf = Math::Prime::Util::MemFree->new;


  # Random primes
  my($rand_prime);
  $rand_prime = random_prime(1000);        # random prime <= limit
  $rand_prime = random_prime(100, 10000);  # random prime within a range
  $rand_prime = random_ndigit_prime(6);    # random 6-digit prime
  $rand_prime = random_nbit_prime(128);    # random 128-bit prime
  $rand_prime = random_safe_prime(192);    # random 192-bit safe prime
  $rand_prime = random_strong_prime(256);  # random 256-bit strong prime
  $rand_prime = random_maurer_prime(256);  # random 256-bit provable prime
  $rand_prime = random_shawe_taylor_prime(256);  # as above


=head1 DESCRIPTION

A module for number theory in Perl.  This includes prime sieving, primality
tests, primality proofs, integer factoring, counts / bounds / approximations
for primes, nth primes, and twin primes, random prime generation,
and much more.

This module is the fastest on CPAN for almost all operations it supports.
This includes
L<Math::Prime::XS>, L<Math::Prime::FastSieve>, L<Math::Factor::XS>,
L<Math::Prime::TiedArray>, L<Math::Big::Factors>, L<Math::Factoring>,
and L<Math::Primality> (when the GMP module is available).
For numbers in the 10-20 digit range, it is often orders of magnitude faster.
Typically it is faster than L<Math::Pari> for 64-bit operations.

All operations support both Perl UV's (32-bit or 64-bit) and bignums.  If
you want high performance with big numbers (larger than Perl's native 32-bit
or 64-bit size), you should install L<Math::Prime::Util::GMP> and
L<Math::BigInt::GMP>.  This will be a recurring theme throughout this
documentation -- while all bignum operations are supported in pure Perl,
most methods will be much slower than the C+GMP alternative.

The module is thread-safe and allows concurrency between Perl threads while
still sharing a prime cache.  It is not itself multi-threaded.  See the
L<Limitations|/"LIMITATIONS"> section if you are using Win32 and threads in
your program.  Also note that L<Math::Pari> is not thread-safe (and will
crash as soon as it is loaded in threads), so if you use
L<Math::BigInt::Pari> rather than L<Math::BigInt::GMP> or the
default backend, things will go pear-shaped.

Two scripts are also included and installed by default:

=over 4

=item *

primes.pl displays primes between start and end values or expressions,
with many options for filtering (e.g. twin, safe, circular, good, lucky,
etc.).  Use C<--help> to see all the options.

=item *

factor.pl operates similarly to the GNU C<factor> program.  It supports
bigint and expression inputs.

=back


=head1 ENVIRONMENT VARIABLES

There are two environment variables that affect operation.  These are
typically used for validation of the different methods or to simulate
systems that have different support.

=head2 MPU_NO_XS

If set to C<1>, everything is run in pure Perl.  No C functions
are loaded or used, as XSLoader is not even called.  All top-level
XS functions are replaced by a pure Perl layer (the PPFE.pm module
that supplies a "Pure Perl Front End").

Caveat: This does not change whether the GMP backend is used.
For as much pure Perl as possible, you will need to set both variables.

If this variable is not set or set to anything other than C<1>, the
module operates normally.

=head2 MPU_NO_GMP

If set to C<1>, the L<Math::Prime::Util::GMP> backend is not
loaded, and operation will be exactly as if it was not installed.

If this variable is not set or set to anything other than C<1>, the
module operates normally.

=head2 MPU_DEVNAMES

If set to C<1>, the PP package will be loaded on startup rather than
on demand, and the package aliases C<MPU>, C<PP>, C<GMP> will be used
for the main, Perl, and GMP packages respectively.
Normally you wouldn't want this both for aggressive namespace pollution
and for performance (there is often no need to load the huge PP module).
But it is convenient if one wants to call the different paths explicitly.

Regarding performance, on a 2020 Macbook M1, normal startup time is
about 10 milliseconds.  With this option set it becomes 45 milliseconds.
This is the reason the PP code is only loaded if needed.
For many purposes this amount of time is trivial, but slower computers
or more time critical short applications will care.


=head1 BIGNUM SUPPORT

By default all functions support bigints.  For performance, you should
install L<Math::Prime::Util::GMP> which will be automatically used as a
backend.

The default bigint class is L<Math::BigInt>, which is not particularly speedy
but is available by default in all Perl distributions, and is well tested.
If you want to try something different, you can install and use L<Math::GMPz>
or L<Math::GMP> which will be B<much> faster.  You can have this module
use and return them using, for example:

  prime_set_config(bigint => Math::GMPz);
  my $n = next_prime(~0);
  say "$n ",ref($n);
  # 18446744073709551629 Math::GMPz

If you use Math::BigInt, I highly recommend also installing one of
L<Math::BigInt::GMPz>, L<Math::BigInt::GMP>, or L<Math::BigInt::LTM>.

If you are using bigints, here are some performance suggestions:

=over 4

=item *

Install a recent version of L<Math::Prime::Util::GMP>, as that will vastly
increase the speed of many of the functions.  This does require the
L<GMP|http://gmplib.org> library be installed on your system, but this
increasingly comes pre-installed or easily available using the OS vendor
package installation tool.

=item *

Install and use L<Math::BigInt::GMP> (or C<GMPz> or C<LTM>), then use
C<use bigint try =E<gt> 'GMP,GMPz,LTM,Pari'> in your script, or on the
command line e.g. C<-Mbigint=lib,GMP>.  Large modular exponentiation is
much faster using the better backends, as are the math and approximation
functions when called with very large inputs.

=item *

I have run these functions on many versions of Perl, and my experience is that
if you're using anything older than Perl 5.14, I would recommend you upgrade
if you are using bignums a lot.  There are some brittle behaviors on 5.12.4
and earlier with bignums.  For example, the default BigInt backend in older
versions of Perl will sometimes convert small results to doubles, resulting
in corrupted output.

=back


=head1 PRIMALITY TESTING

This module provides three functions for general primality testing, as
well as numerous specialized functions.  The three main functions are:
L</is_prob_prime> and L</is_prime> for general use, and L</is_provable_prime>
for proofs.  For inputs below C<2^64> the functions are identical and
fast deterministic testing is performed.  That is, the results will always
be correct and should take at most a few microseconds for any input.  This
is hundreds to thousands of times faster than other CPAN modules.  For
inputs larger than C<2^64>, an extra-strong
L<BPSW test|http://en.wikipedia.org/wiki/Baillie-PSW_primality_test>
is used.  See the L</PRIMALITY TESTING NOTES> section for more
discussion.

Following the semantics used by Pari/GP, all primality test functions
allow a negative primary argument, but will return false.
All inputs must be integers or an error is raised.


=head1 FUNCTIONS

=head2 is_prime

  print "$n is prime" if is_prime($n);

Given an integer C<n>, returns 0 if the number is composite,
1 if it is probably prime, and 2 if it is definitely prime.
For numbers smaller than C<2^64> it will only
return 0 (composite) or 2 (definitely prime), as this range has been
exhaustively tested and has no counterexamples.
For larger numbers, an extra-strong BPSW test is used.
If L<Math::Prime::Util::GMP> is installed, some additional primality tests
are also performed, and a quick attempt is made to perform a primality
proof, so it will return 2 for many other inputs.

Also see the L</is_prob_prime> function, which will never do additional
tests, and the L</is_provable_prime> function which will construct a proof
that the input is prime and returns 2 for almost all primes (at the
expense of speed).

For native precision numbers (anything smaller than C<2^64>, all three
functions are identical and use a deterministic set of tests (selected
Miller-Rabin bases or BPSW).  For larger inputs both L</is_prob_prime> and
L</is_prime> return probable prime results using the extra-strong
Baillie-PSW test, which has had no counterexample found since it was
published in 1980.

For cryptographic key generation, you may want even more testing for probable
primes (NIST recommends some additional M-R tests).  This can be done using
a different test (e.g. L</is_frobenius_underwood_pseudoprime>) or using
additional M-R tests with random bases with L</miller_rabin_random>.
Even better, make sure L<Math::Prime::Util::GMP> is installed and use
L</is_provable_prime> which should be reasonably fast for sizes under
2048 bits.  Another possibility is to use
L<Math::Prime::Util/random_maurer_prime> or
L<Math::Prime::Util/random_shawe_taylor_prime> which construct random
provable primes.


=head2 primes

Returns all the primes between the lower and upper limits (inclusive), with
a lower limit of C<2> if none is given.

An array reference is returned (with large lists this is much faster and uses
less memory than returning an array directly).

  my $aref1 = primes( 1_000_000 );
  my $aref2 = primes( 1_000_000_000_000, 1_000_000_001_000 );

  my @primes = @{ primes( 500 ) };

  print "$_\n" for @{primes(20,100)};

Sieving will be done if required.  The algorithm used will depend on the range
and whether a sieve result already exists.  Possibilities include primality
testing (for very small ranges), a Sieve of Eratosthenes using wheel
factorization, or a segmented sieve.


=head2 next_prime

  $n = next_prime($n);

Returns the next prime greater than the input number.  The result will be a
bigint if it can not be exactly represented in the native int type
(larger than C<4,294,967,291> in 32-bit Perl;
larger than C<18,446,744,073,709,551,557> in 64-bit).


=head2 prev_prime

  $n = prev_prime($n);

Returns the prime preceding the input number (i.e. the largest prime that is
strictly less than the input).  C<undef> is returned if the input is C<2>
or lower.

The behavior in various programs of the I<previous prime> function is varied.
Pari/GP and L<Math::Pari> returns the input if it is prime, as does
L<Math::Prime::FastSieve/nearest_le>.  When given an input such that the
return value will be the first prime less than C<2>,
L<Math::Prime::FastSieve>, L<Math::Pari>, Pari/GP, and older versions of
MPU will return C<0>.  L<Math::Primality> and the current MPU will return
C<undef>.  WolframAlpha returns C<-2>.  Maple gives a range error.



=head2 forprimes

  forprimes { say } 100,200;                  # print primes from 100 to 200

  $sum=0;  forprimes { $sum += $_ } 100000;   # sum primes to 100k

  forprimes { say if is_prime($_+2) } 10000;  # print twin primes to 10k

Given a block and either an end count or a start and end pair, calls the
block for each prime in the range.  Compared to getting a big array of primes
and iterating through it, this is more memory efficient and perhaps more
convenient.  This will almost always be the fastest way to loop over a range
of primes.  Nesting and use in threads are allowed.

Math::BigInt objects may be used for the range.

For some uses an iterator (L</prime_iterator>, L</prime_iterator_object>)
or a tied array (L<Math::Prime::Util::PrimeArray>) may be more convenient.
Objects can be passed to functions, and allow early loop exits.


=head2 forcomposites

  forcomposites { say } 1000;
  forcomposites { say } 2000,2020;

Given a block and either an end number or a start and end pair, calls the
block for each composite in the inclusive range.  The composites,
L<OEIS A002808|http://oeis.org/A002808>, are the numbers greater than 1
which are not prime:  C<4, 6, 8, 9, 10, 12, 14, 15, ...>.


=head2 foroddcomposites

Similar to L</forcomposites>, but skipping all even numbers.
The odd composites, L<OEIS A071904|http://oeis.org/A071904>, are the
numbers greater than 1 which are not prime and not divisible by two:
C<9, 15, 21, 25, 27, 33, 35, ...>.


=head2 forsemiprimes

Similar to L</forcomposites>, but only giving composites with exactly
two factors.
The semiprimes, L<OEIS A001358|http://oeis.org/A001358>, are the
products of two primes:
C<4, 6, 9, 10, 14, 15, 21, 22, 25, ...>.

This is essentially equivalent to:

  forcomposites { if (is_semiprime($_)) { ... } }

=head2 foralmostprimes

  foralmostprimes  { say }  3,  1000,2000;  # 3-almost-primes in [1000,2000]

Similar to L</forprimes>, L</forsemiprimes>, etc. but takes an additional
first argument C<k> and loops through the inclusive range for only those
numbers with exactly C<k> factors.  If C<k=1> these are the primes, if
C<k=2> these are the semiprimes, if C<k=3> these are the integers in the
range with exactly 3 prime factors, etc.

This is functionally equivalent to:

  for ($a .. $b) { if (is_almost_prime($k,$_)) { ... } }
  # or
  for ($a .. $b) { if (prime_bigomega($_) == $k) { ... } }

though B<significantly> faster and avoids issues with large loop variables.

=head2 forfactored

  forfactored { say "$_: @_"; } 100;

Given a block and either an end number or start/end pair, calls the block for
each number in the inclusive range.  C<$_> is set to the number while C<@_>
holds the factors.  Especially for small inputs or large ranges, this can be
faster than calling L</factor> on each sequential value.

Similar to the arrays returned by similar functions such as L</forpart>,
the values in C<@_> are read-only.
Any attempt to modify them will result in undefined behavior.

This corresponds to the Pari/GP 2.10 C<forfactored> function.


=head2 forsquarefree

Similar to L</forfactored>, but skipping numbers in the range that have a
repeated factor.  Inside the block, the moebius function can be cheaply
computed as C<((scalar(@_) & 1) ? -1 : 1)> or similar.

This corresponds to the Pari/GP 2.10 C<forsquarefree> function.

=head2 forsquarefreeint

Similar to L</forsquarefree>, but only sieves for square-free integers in
the range (in segments so very large ranges still use little memory).
No factoring information is returned: the C<@_> variable is not set.
In return it is 2 to 20 times faster.

As with range functions such as L</foralmostprimes> this can be B<much>
faster than calling L</is_square_free> for each integer in a large range.


=head2 fordivisors

  fordivisors { $prod *= $_ } $n;

Given a block and a non-negative number C<n>, the block is called with
C<$_> set to each divisor in sorted order.  Also see L</divisor_sum>.


=head2 forpart

  forpart { say "@_" } 25;           # unrestricted partitions
  forpart { say "@_" } 25,{n=>5}     # ... with exactly 5 values
  forpart { say "@_" } 25,{nmax=>5}  # ... with <=5 values

Given a non-negative number C<n>, the block is called with C<@_> set to
the array of additive integer partitions.  The operation is very similar
to the C<forpart> function in Pari/GP 2.6.x, though the ordering is
different.  The ordering is lexicographic.
Use L</partitions> to get just the count of unrestricted partitions.


An optional hash reference may be given to produce restricted partitions.
Each value must be a non-negative integer.  The allowable keys are:

  n       restrict to exactly this many values
  amin    all elements must be at least this value
  amax    all elements must be at most this value
  nmin    the array must have at least this many values
  nmax    the array must have at most this many values
  prime   all elements must be prime (non-zero) or non-prime (zero)

Like forcomb and forperm, the partition return values are read-only.  Any
attempt to modify them will result in undefined behavior.


=head2 forcomp

Similar to L</forpart>, but iterates over integer compositions rather than
partitions.  This can be thought of as all orderings of partitions, or
alternately partitions may be viewed as an ordered subset of compositions.
The ordering is lexicographic.  All options from L</forpart> may be used.

The number of unrestricted compositions of C<n> is C<2^(n-1)>.

=head2 forcomb

Given non-negative arguments C<n> and C<k>, the block is called with C<@_>
set to the C<k> element array of values from C<0> to C<n-1> representing
the combinations in lexicographical order.  While the L</binomial> function
gives the total number, this function can be used to enumerate the choices.

Rather than give a data array as input, an integer is used for C<n>.
A convenient way to map to array elements is:

  forcomb { say "@data[@_]" } @data, 3;

where the block maps the combination array C<@_> to array values, the
argument for C<n> is given the array since it will be evaluated as a scalar
and hence give the size, and the argument for C<k> is the desired size of
the combinations.

Like forpart and forperm, the index return values are read-only.  Any
attempt to modify them will result in undefined behavior.

If the second argument C<k> is not supplied, then all k-subsets are returned
starting with the smallest set C<k=0> and continuing to C<k=n>.  Each
k-subset is in lexicographical order.  This is the power set of C<n>.

This corresponds to the Pari/GP 2.10 C<forsubset> function.


=head2 forperm

Given non-negative argument C<n>, the block is called with C<@_> set to
the C<k> element array of values from C<0> to C<n-1> representing
permutations in lexicographical order.
The total number of calls will be C<n!>.

Rather than give a data array as input, an integer is used for C<n>.
A convenient way to map to array elements is:

  forperm { say "@data[@_]" } @data;

where the block maps the permutation array C<@_> to array values, and the
argument for C<n> is given the array since it will be evaluated as a scalar
and hence give the size.

Like forpart and forcomb, the index return values are read-only.  Any
attempt to modify them will result in undefined behavior.


=head2 forderange

Similar to forperm, but iterates over derangements.  This is the set of
permutations skipping any which maps an element to its original position.


=head2 formultiperm

  # Show all anagrams of 'serpent':
  formultiperm { say join("",@_) } [split(//,"serpent")];

Similar to L</forperm> but takes an array reference as an argument.  This
is treated as a multiset, and the block will be called with each multiset
permutation.  While the standard permutation iterator takes a scalar and
returns index permutations, this takes the set itself.

If all values are unique, then the results will be the same as a standard
permutation.  Otherwise, the results will be similar to a standard
permutation removing duplicate entries.  While generating all
permutations and filtering out duplicates works, it is very slow for large
sets.  This iterator will be much more efficient.

There is no ordering requirement for the input array reference.  The results
will be in lexicographic order.


=head2 forsetproduct

  forsetproduct { say "@_" } [1,2,3],[qw/a b c/],[qw/@ $ !/];

Takes zero or more array references as arguments and iterates over the
set product (i.e. Cartesian product or cross product) of the lists.
The given subroutine is repeatedly called with C<@_> set to the
current list.
Since no de-duplication is done, this is not literally a C<set> product.

While zero or one array references are valid, the result is not very
interesting.  If any array reference is empty, the product is
empty, so no subroutine calls are performed.

The subroutine is given an array whose values are aliased to the
inputs, and are I<not> set to read-only.  Hence modifying the array
inside the subroutine will cause side-effects.

As with other iterators, the C<lastfor> function will cause an early exit.


=head2 lastfor

  forprimes { lastfor,return if $_ > 1000; $sum += $_; } 1e9;

Calling lastfor requests that the current for... loop stop after this
call.  Ideally this would act exactly like a C<last> inside a loop,
but technical reasons mean it does not exit the block early, hence
one typically adds a C<return> if needed.


=head2 prime_iterator


  my $it = prime_iterator;
  $sum += $it->() for 1..100000;

Returns a closure-style iterator.  The start value defaults to the first
prime (2) but an initial value may be given as an argument, which will result
in the first value returned being the next prime greater than or equal to the
argument.  For example, this:

  my $it = prime_iterator(200);  say $it->();  say $it->();

will return 211 followed by 223, as those are the next primes E<gt>= 200.
On each call, the iterator returns the current value and increments to
the next prime.

Other options include L</forprimes> (more efficiency, less flexibility),
L<Math::Prime::Util::PrimeIterator> (an iterator with more functionality),
or L<Math::Prime::Util::PrimeArray> (a tied array).


=head2 prime_iterator_object

  my $it = prime_iterator_object;
  while ($it->value < 100) { say $it->value; $it->next; }
  $sum += $it->iterate for 1..100000;

Returns a L<Math::Prime::Util::PrimeIterator> object.  A shortcut that loads
the package if needed, calls new, and returns the object.  See the
documentation for that package for details.  This object has more features
than the simple one above (e.g. the iterator is bi-directional), and also
handles iterating across bigints.


=head2 prime_count

  my $primepi = prime_count( 1_000 );
  my $pirange = prime_count( 1_000, 10_000 );

Returns the Prime Count function C<Pi(n)>, also called C<primepi> in some
math packages.  When given two arguments, it returns the inclusive
count of primes between the ranges.  E.g. C<(13,17)> returns 2, C<(14,17)>
and C<(13,16)> return 1, C<(14,16)> returns 0.

The current implementation decides based on the ranges whether to use a
segmented sieve with a fast bit count, or the extended LMO algorithm.
The former is preferred for small sizes as well as small ranges.
The latter is much faster for large ranges.

The segmented sieve is very memory efficient and is quite fast even with
large base values.  Its complexity is approximately C<O(sqrt(a) + (b-a))>,
where the first term is typically negligible below C<~ 10^11>.  Memory use
is proportional only to C<sqrt(a)>, with total memory use under 1MB for any
base under C<10^14>.

The extended LMO method has complexity approximately
C<O(b^(2/3)) + O(a^(2/3))>, and also uses low memory.
A calculation of C<Pi(10^14)> completes in a few seconds, C<Pi(10^15)>
in well under a minute, and C<Pi(10^16)> in about one minute.  In
contrast, even parallel primesieve would take over a week on a
similar machine to determine C<Pi(10^16)>.

Also see the function L</prime_count_approx> which gives a very good
approximation to the prime count, and L</prime_count_lower> and
L</prime_count_upper> which give tight bounds to the actual prime count.
These functions return quickly for any input, including bigints.


=head2 prime_count_upper

Returns a proven upper bound on the number of primes up to C<n>.
See L</prime_count_lower> for details common to both functions.

=head2 prime_count_lower

  my $lower_limit = prime_count_lower($n);
  my $upper_limit = prime_count_upper($n);
  #   $lower_limit  <=  prime_count(n)  <=  $upper_limit

Returns a proven lower bound on the number of primes up to C<n>.
These are analytical routines, so will take a fixed amount of time and no
memory.  The actual C<prime_count> will always be equal to or between these
numbers.

A common place these would be used is sizing an array to hold the first C<$n>
primes.  It may be desirable to use a bit more memory than is necessary, to
avoid calling C<prime_count>.

These routines use verified tight limits below a range at least C<2^35>.
For larger inputs various methods are used including Dusart (2010),
Büthe (2014,2015), and Axler (2014).
These bounds do not assume the Riemann Hypothesis.
If the configuration option C<assume_rh> has been set (it is off by default),
then the Schoenfeld (1976) bounds can be used for very large values.


=head2 prime_count_approx

  print "there are about ",
        prime_count_approx( 10 ** 18 ),
        " primes below one quintillion.\n";

Returns an approximation to the C<prime_count> function, without having to
generate any primes.  For values under C<10^36> this uses the Riemann R
function, which is quite accurate: an error of less than C<0.0005%> is typical
for input values over C<2^32>, and decreases as the input gets larger.

A slightly faster but much less accurate answer can be obtained by averaging
the upper and lower bounds.


=head2 is_prime_power

Given an integer C<n>, returns C<k> if C<n = p^k> for some prime p,
and zero otherwise.

If a second argument is present, it must be a scalar reference.  If the
return value is non-zero, then it will be set to C<p>.

This corresponds to Pari/GP's C<isprimepower> function.  It is related to
Mathematica's C<PrimePowerQ[n]> function.
These all return zero/false for C<n=1>.

This is the L<OEIS series A246655|http://oeis.org/A246655>.

=head2 prime_powers

  my $aref = prime_powers( 10**4 );

Given either two non-negative limits C<lo>, C<hi>, or one non-negative
limit C<hi>, returns an array reference with all prime powers between
the limits (inclusive).  With only one input, the lower limit is C<2>.

The array reference values will be all C<p^e> where
C<< lo <= p^e <= hi >> with C<p> prime and C<< e >= 1 >>.  Hence this
includes the primes as well as higher powers of primes.

See also L</primes> and L</prime_power_count>.

=head2 next_prime_power

Given an integer C<n>, returns the smallest prime power greater than C<|n|>.
Similar to L</next_prime>, but also includes powers of primes.

=head2 prev_prime_power

Given an integer C<n>, returns the greatest prime power less than C<|n|>.
Similar to L</prev_prime>, but also includes powers of primes.
If given C<|n|> less than 3, C<undef> will be returned.

=head2 prime_power_count

Given a single non-negative integer C<n>, returns the count of
prime powers less than or equal to C<n>.
If given two non-negative integers C<lo> and C<hi>, returns the count
of prime powers between C<lo> and C<hi> inclusive.

These are prime powers with exponent greater than 0.
I.e. the prime powers not including C<1>.
This is L<OEIS series A025528|http://oeis.org/A025528>.

=head2 prime_power_count_approx

Given a non-negative integer C<n>, quickly returns a
good estimate of the count of prime powers less than or equal to C<n>.

=head2 prime_power_count_lower

Given a non-negative integer C<n>, quickly returns a
lower bound of the count of prime powers less than or equal to C<n>.
The actual count will always be greater than or equal to the result.

=head2 prime_power_count_upper

Given a non-negative integer C<n>, quickly returns an
upper bound of the count of prime powers less than or equal to C<n>.
The actual count will always be less than or equal to the result.

=head2 nth_prime_power

Given a non-negative integer C<n>, returns the C<n>-th prime power.

=head2 nth_prime_power_approx

Given a non-negative integer C<n>, quickly returns a
good estimate of the C<n>-th prime power.

=head2 nth_prime_power_lower

Given a non-negative integer C<n>, quickly returns a
lower bound of the C<n>-th prime power.
The actual value will always be greater than or equal to the result.

=head2 nth_prime_power_upper

Given a non-negative integer C<n>, quickly returns an
upper bound of the C<n>-th prime power.
The actual value will always be less than or equal to the result.



=head2 twin_primes

Returns the lesser of twin primes between the lower and upper limits
(inclusive), with a lower limit of C<2> if none is given.  This is
L<OEIS A001359|http://oeis.org/A001359>.
Given a twin prime pair C<(p,q)> with C<q = p + 2>, C<p> prime,
and C<q> prime, this function uses C<p> to represent the pair.  Hence the
bounds need to include C<p>, and the returned list will have C<p> but not C<q>.

This works just like the L</primes> function, though only the first primes of
twin prime pairs are returned.  Like that function, an array reference is
returned.


=head2 twin_prime_count

Similar to prime count, but returns the count of twin primes (primes C<p>
where C<p+2> is also prime).  Takes either a single number indicating a count
from 2 to the argument, or two numbers indicating a range.

The primes being counted are the first value, so a range of C<(3,5)> will
return a count of two, because both C<3> and C<5> are counted as twin primes.
A range of C<(12,13)> will return a count of zero, because neither C<12+2>
nor C<13+2> are prime.  In contrast, C<primesieve> requires all elements of
a constellation to be within the range to be counted, so would return one for
the first example (C<5> is not counted because its pair C<7> is not in the
range).

There is no useful formula known for this, unlike prime counts.  We sieve
for the answer, using some small table acceleration.

=head2 twin_prime_count_approx

Returns an approximation to the twin prime count of C<n>.  This returns
quickly and has a very small error for large values.  The method used is
conjecture B of Hardy and Littlewood 1922, as stated in
Sebah and Gourdon 2002.  For inputs under 10M, a correction factor is
additionally applied to reduce the mean squared error.


=head2 semi_primes

Returns an array reference to semiprimes between the lower and upper
limits (inclusive), with a lower limit of C<4> if none is given.
This is L<OEIS A001358|http://oeis.org/A001358>.
The semiprimes are composite integers which are products of
exactly two primes.

This works just like the L</primes> function.
Like that function, an array reference is returned.

=head2 semiprime_count

Similar to prime count, but returns the count of semiprimes (composites with
exactly two factors).  Takes either a single number indicating a count
from 2 to the argument, or two numbers indicating a range.

A fast method that requires computation only to the square root of the
range end is used, unless the range is so small that walking it is faster.

=head2 semiprime_count_approx

Returns an approximation to the semiprime count of C<n>.
This returns quickly and is square root accurate for native size inputs.

The series of Crisan and Erban (2020) is used with a maximum of 19 terms.
Truncation is performed at empirical good crossovers.  Clamping is done
as needed at crossovers to ensure monotonic results.


=head2 almost_primes

  my $ref_to_3_almost_primes = almost_primes(3, 1000, 2000);

Takes a non-negative integer argument C<k> and either one or two additional
non-negative integer arguments indicating the upper limit or lower and upper
limits.  The limits are inclusive.
The k-almost-primes are integers which have exactly C<k> prime factors.

This works just like the L</primes> function.
Like that function, an array reference is returned.

With C<k=1> these are the primes (L<OEIS A000040|http://oeis.org/A000040>).
With C<k=2> these are the semiprimes (L<OEIS A001358|http://oeis.org/A001358>).
With C<k=3> these are the 3-almost-primes (L<OEIS A014612|http://oeis.org/A014612>).
With C<k=4> these are the 4-almost-primes (L<OEIS A014613|http://oeis.org/A014613>).
OEIS sequences can be found through C<k=20>.

=head2 almost_prime_count

  say almost_prime_count(3,10000); # number of 3-almost-primes <= 10000

Given non-negative integers C<k> and C<n>, returns the count of
C<k>-almost-prime numbers up to and including C<n>.  With C<k=1> this
is the standard prime count.  With C<k=2> this is the semiprime count.
In general, this is the count of all integers through C<n> that have
exactly C<k> prime factors.

The implementation uses nested prime count sums, and caching along
with LMO prime counts to get quite reasonable speeds.

=head2 almost_prime_count_approx

A fast approximation of the C<k>-almost-prime count of C<n>.

The current implementation for C<n> greater than 64-bit is not well tested.

=head2 almost_prime_count_lower

Quickly returns a lower bound for the C<k>-almost-prime count of C<n>.
The actual count will be greater than or equal to this result.

The current implementation for C<n> greater than 64-bit is not well tested.

=head2 almost_prime_count_upper

Quickly returns an upper bound for the C<k>-almost-prime count of C<n>.
The actual count will be less than or equal to this result.

The current implementation for C<n> greater than 64-bit is not well tested.


=head2 omega_primes

Takes a non-negative integer argument C<k> and either one or two additional
non-negative integer arguments indicating the upper limit or lower and upper
limits.  The limits are inclusive.
The k-omega-primes are positive integers which have exactly C<k> distinct
prime factors, with possible multiplicity.  Hence these numbers are divisible
by exactly C<k> different primes.

The k-omega-primes (not a common term) are exactly those integers where
C<prime_omega(n) == k>.
Compare to k-almost-primes where C<prime_bigomega(n) == k>.

With C<k=1> these are the prime powers.
With C<k=2> these are L<OEIS A007774|http://oeis.org/A007774>.
With C<k=3> these are L<OEIS A033992|http://oeis.org/A033992>.

=head2 omega_prime_count

Given non-negative integers C<k> and C<n>, returns the count of
C<k>-omega-prime numbers from C<1> up to and including C<n>.
This is the count of all positive integers through C<n> that are
divisible by exactly C<k> different primes.

The implementation uses nested loops over prime powers.

Though we have defined C<prime_omega(0) = 1>, it is not included.

=head2 ramanujan_primes

Returns the Ramanujan primes R_n between the upper and lower limits
(inclusive), with a lower limit of C<2> if none is given.  This is
L<OEIS A104272|http://oeis.org/A104272>.  These are the Rn such that if
C<< x > Rn >> then L</prime_count>(n) - L</prime_count>(n/2) C<< >= n >>.

This has a similar API to the L</primes> and L</twin_primes> functions, and
like them, returns an array reference.

Generating Ramanujan primes takes some effort, including overhead to cover
a range.  This will be substantially slower than generating standard primes.

=head2 ramanujan_prime_count

Similar to prime count, but returns the count of Ramanujan primes.  Takes
either a single number indicating a count from 2 to the argument, or
two numbers indicating a range.

While not nearly as efficient as L</prime_count>, this does use a number of
speedups that result in it being much more efficient than generating all
the Ramanujan primes.

=head2 ramanujan_prime_count_approx

A fast approximation of the count of Ramanujan primes under C<n>.

=head2 ramanujan_prime_count_lower

A fast lower limit on the count of Ramanujan primes under C<n>.

=head2 ramanujan_prime_count_upper

A fast upper limit on the count of Ramanujan primes under C<n>.


=head2 sieve_range

  my @candidates = sieve_range(2**1000, 10000, 40000);

Given a start value C<n>, and native unsigned integers C<width> and C<depth>,
a sieve of maximum depth C<depth> is done for the C<width> consecutive
numbers beginning with C<n>.  An array of offsets from the start is returned.

The returned list contains those offsets in the range C<n> to C<n+width-1>
where C<n + offset> has no prime factors smaller than itself and
less than or equal to C<depth>.  Hence a depth of 2 will remove all even
numbers (other than 2 itself if it is in the range).
A depth of 3 will remove all numbers divisible by 2 or 3 other than those
primes themselves.


=head2 sieve_prime_cluster

  my @s = sieve_prime_cluster(1, 1e9, 2,6,8,12,18,20);

Efficiently finds prime clusters between the first two arguments C<low>
and C<high>.  The remaining arguments describe the cluster.
The cluster values must be even, less than 31 bits, and strictly increasing.
Given a cluster set C<C>, the returned values are all primes in the
range where C<p+c> is prime for each C<c> in the cluster set C<C>.
For returned values under C<2^64>, all cluster values are
definitely prime.  Above this range, all cluster values are BPSW
probable primes (no counterexamples known).

This function returns an array rather than an array reference.
Typically the number of returned values is much lower than for
other primes functions, so this uses the more convenient array
return.  This function has an identical signature to the function
of the same name in L<Math::Prime::Util:GMP>.

The cluster is described as offsets from 0, with the implicit prime
at 0.  Hence an empty list is asking for all primes (the cluster
C<p+0>).  A list with the single value C<2> will find all twin primes
(the cluster where C<p+0> and C<p+2> are prime).  The list C<2,6,8>
will find prime quadruplets.  Note that there is no requirement that
the list denote a constellation (a cluster with minimal distance) --
the list C<42,92,606> is just fine.

=head2 sum_primes

Returns the summation of primes between the lower and upper limits
(inclusive), with a lower limit of C<2> if none is given.  This is
essentially similar to either of:

    $sum = 0; forprimes { $sum += $_ } $low,$high;  $sum;
    # or
    vecsum( @{ primes($low,$high) } );

but is much more efficient.

The current implementation is a mix of small-table-enhanced sieve count
for sums that fit in a UV, an efficient sieve count for small ranges, and
a Legendre sum method, including XS support for 128-bit results.

While this is fairly efficient, the state of the art is Kim Walisch's
L<primesum|https://github.com/kimwalisch/primesum>.
It is recommended for very large values, as it can be hundreds of times
faster.

=head2 print_primes

  print_primes(1_000_000);             # print the first 1 million primes
  print_primes(1000, 2000);            # print primes in range
  print_primes(2,1000,fileno(STDERR))  # print to a different descriptor

With a single argument this prints all primes from 2 to C<n> to standard
out.  With two arguments it prints primes between C<low> and C<high> to
standard output.  With three arguments it prints primes between C<low>
and C<high> to the file descriptor given.  If the file descriptor cannot
be written to, this will croak with "print_primes write error".  It will
produce identical output to:

    forprimes { say } $low,$high;

The point of this function is just efficiency.  It is over 10x faster
than using C<say>, C<print>, or C<printf>, though much more limited
in functionality.  A later version may allow a file handle as the third
argument.


=head2 nth_prime

  say "The ten thousandth prime is ", nth_prime(10_000);

Returns the prime that lies in index C<n> in the array of prime numbers.  Put
another way, this returns the smallest C<p> such that C<< Pi(p) >= n >>.

Like most programs with similar functionality, this is one-based.
C<nth_prime(0)> returns C<undef>, C<nth_prime(1)> returns C<2>.

For relatively small inputs (below 1 million or so), this does a sieve over
a range containing the nth prime, then counts up to the number.  This is fairly
efficient in time and memory.  For larger values, create a low-biased estimate
using the inverse logarithmic integral, use a fast prime count, then sieve in
the small difference.

While this method is thousands of times faster than generating primes, and
doesn't involve big tables of precomputed values, it still can take a fair
amount of time for large inputs.  Calculating the C<10^12th> prime takes
about 1 second, the C<10^13th> prime takes under 10 seconds, and the
C<10^14th> prime (3475385758524527) takes under 30 seconds.  Think about
whether a bound or approximation would be acceptable, as they can be
computed analytically.

If the result is larger than a native integer size (32-bit or 64-bit), the
result will take a very long time.  A later version of
L<Math::Prime::Util::GMP> may include this functionality which would help for
32-bit machines.


=head2 nth_prime_upper

Returns a proven upper bound on the Nth prime.
See L</nth_prime_lower> for details common to both functions.

=head2 nth_prime_lower

  my $lower_limit = nth_prime_lower($n);
  my $upper_limit = nth_prime_upper($n);
  # For all $n:   $lower_limit  <=  nth_prime($n)  <=  $upper_limit

Returns a proven lower bound on the Nth prime.  No sieving is
done, so these are fast even for large inputs.

For tiny values of C<n>. exact answers are returned.  For small inputs, an
inverse of the opposite prime count bound is used.  For larger values, the
Dusart (2010) and Axler (2013) bounds are used.


=head2 nth_prime_approx

  say "The one trillionth prime is ~ ", nth_prime_approx(10**12);

Returns an approximation to the C<nth_prime> function, without having to
generate any primes.  For values where the nth prime is smaller than
C<2^64>, the inverse Riemann R function is used.  For larger values,
the inverse logarithmic integral is used.

The value returned will not necessarily be prime.  This applies to all
the following nth prime approximations, where the returned value is
close to the real value, but no effort is made to coerce the result
to the nearest set element.


=head2 nth_twin_prime

Returns the Nth twin prime.  This is done via sieving and counting, so
is not very fast for large values.

=head2 nth_twin_prime_approx

Returns an approximation to the Nth twin prime.  A curve fit is used for
small inputs (under 1200), while for larger inputs a binary search is done
on the approximate twin prime count.

=head2 nth_semiprime

Returns the Nth semiprime, similar to where a C<forsemiprimes> loop would
end after C<N> iterations, but much more efficiently.

=head2 nth_semiprime_approx

Returns an approximation to the Nth semiprime.  The approximation is
orders of magnitude better than the simple C<n log n / log log n>
approximation for large C<n>.  E.g. for C<n=10^12> the simple estimate
is within 3.6%, but this function is within 0.000012%.

=head2 nth_almost_prime

  say "500th number with exactly 3 factors: ", nth_almost_prime(3,500);

A C<k>-almost prime is a product of C<k> prime numbers,
counted with multiplicity.  That is, there are exactly C<k> prime
factors (which do not have to be distinct from each other).

Given non-negative integers C<k> and C<n>, returns the
C<n>-th C<k>-almost prime.
With C<k=1> this is the nth prime.
With C<k=2> this is the nth semiprime.

The implementation does a binary search lookup with
L</almost_prime_count> so is reasonably efficient for large values.

C<undef> is returned for C<n == 0> and for all C<k == 0>
other than C<n == 1>.

=head2 nth_almost_prime_approx

A fast approximation of the C<n>-th C<k>-almost prime.

=head2 nth_almost_prime_lower

Quickly returns a lower bound for the C<n>-th C<k>-almost prime.
The actual nth k-almost-prime will be greater than or equal to this result.

=head2 nth_almost_prime_upper

Quickly returns an upper bound for the C<n>-th C<k>-almost prime.
The actual nth k-almost-prime will be less than or equal to this result.


=head2 nth_omega_prime

Given non-negative integers C<k> and C<n>, returns the
C<n>-th C<k>-omega prime.
This is the C<n>-th integer divisible by exactly C<k> different primes.

The implementation does a binary search lookup with
L</omega_prime_count> so is reasonably efficient for large values.

C<undef> is returned for C<n == 0> and for all C<k == 0>
other than C<n == 1>.


=head2 nth_ramanujan_prime

Returns the Nth Ramanujan prime.  For reasonable size values of C<n>, e.g.
under C<10^8> or so, this is relatively efficient for single calls.  If
multiple calls are being made, it will be much more efficient to get the
list once.

=head2 nth_ramanujan_prime_approx

A fast approximation of the Nth Ramanujan prime.

=head2 nth_ramanujan_prime_lower

A fast lower limit on the Nth Ramanujan prime.

=head2 nth_ramanujan_prime_upper

A fast upper limit on the Nth Ramanujan prime.



=head2 is_pseudoprime

Given an integer C<n> and zero or more positive bases,
returns 1 if C<n> is positive and a probable prime to each base,
and returns 0 otherwise.
This is the simple Fermat primality test.
Removing primes, given base 2 this produces the sequence L<OEIS A001567|http://oeis.org/A001567>.

If no bases are given, base 2 is used.  All bases must be 2 or greater.

For practical use, L</is_strong_pseudoprime> is a much stronger test with
similar or better performance.

Note that there is a set of composites (the Carmichael numbers) that will
pass this test for all bases.  This downside is not shared by the Euler
and strong probable prime tests (also called the Solovay-Strassen
and Miller-Rabin tests).

=head2 is_euler_pseudoprime

Given an integer C<n> and zero or more positive bases,
returns 1 if C<n> is positive and an Euler probable prime to each base,
and returns 0 otherwise.
This is the Euler test, sometimes called the Euler-Jacobi test.
Removing primes, given base 2 this produces the sequence L<OEIS A047713|http://oeis.org/A047713>.

If no bases are given, base 2 is used.  All bases must be 2 or greater.

If 0 is returned, then the number really is a composite (for bases less than n).
If 1 is returned, then it is either a prime or an Euler pseudoprime to all the given bases.
Given enough distinct bases, the chances become very high that the
number is actually prime.

This test forms the basis of the Solovay-Strassen test, which is a precursor
to the Miller-Rabin test (which uses the strong probable prime test).  There
are no analogies to the Carmichael numbers for this test.
For the Euler test, at I<most> 1/2 of witnesses pass for a composite, while
at most 1/4 pass for the strong pseudoprime test.

=head2 is_strong_pseudoprime

  my $maybe_prime = is_strong_pseudoprime($n);
  my $probably_prime = is_strong_pseudoprime($n, 2, 3, 5, 7, 11, 13, 17);

Given an integer C<n> and zero or more positive bases,
returns 1 if C<n> is positive and a strong probable prime to each base,
and returns 0 otherwise.

If no bases are given, base 2 is used.  All bases must be 2 or greater.

If 0 is returned, then the number really is a composite (for any base).
If 1 is returned, then it is either a prime or a strong pseudoprime to all the given bases.
Given enough distinct bases, the chances become very high that the
number is actually prime.

This is usually used in combination with other tests to make either stronger
tests (e.g. the strong BPSW test) or deterministic results for numbers less
than some verified limit (e.g. it has long been known that no more than three
selected bases are required to give correct primality test results for any
32-bit number).  Given the small chances of passing multiple bases, there
are some math packages that just use multiple MR tests for primality testing.

Even inputs other than 2 will always return 0 (composite).  While the
algorithm does run with even input, most sources define it only on odd input.
Returning composite for all non-2 even input makes the function match most
other implementations including L<Math::Primality>'s C<is_strong_pseudoprime>
function.

Generally, bases of interest are between C<2> and C<n-2>.
Bases C<1> and C<n-1> will return 1 for any odd composites.
Most sources do not define the test for bases equal to C<0 mod n>,
and many do not for any bases larger than C<n>.
We allow all bases, noting that the case C<base = 0 mod n> is defined as 1.
This allows primes to return 1 regardless of the base.

=head2 is_lucas_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and a
standard Lucas probable prime using the Selfridge method of choosing
D, P, and Q (some sources call this a Lucas-Selfridge pseudoprime).
Removing primes, this produces the sequence
L<OEIS A217120|http://oeis.org/A217120>.

=head2 is_strong_lucas_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and a
strong Lucas probable prime using the Selfridge method of choosing
D, P, and Q (some sources call this a strong Lucas-Selfridge pseudoprime).
This is one half
of the BPSW primality test (the Miller-Rabin strong pseudoprime test with
base 2 being the other half).  Removing primes, this produces the sequence
L<OEIS A217255|http://oeis.org/A217255>.

=head2 is_extra_strong_lucas_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and an
extra strong Lucas probable prime as defined in
L<Grantham 2000|http://www.ams.org/mathscinet-getitem?mr=1680879>.
This test has more stringent conditions than the strong Lucas test,
and produces about 60% fewer pseudoprimes.
Performance is typically 20-30% I<faster> than the strong Lucas test.

The parameters are selected using the
L<Baillie-OEIS method|http://oeis.org/A217719>:
increment C<P> from C<3> until C<jacobi(D,n) = -1>.
Removing primes, this produces the sequence
L<OEIS A217719|http://oeis.org/A217719>.

=head2 is_almost_extra_strong_lucas_pseudoprime

This is similar to the L</is_extra_strong_lucas_pseudoprime> function, but
does not calculate C<U>, so is a little faster, but also weaker.
With the current implementations, there is little reason to prefer this unless
trying to reproduce specific results.  The extra-strong implementation has been
optimized to use similar features, removing most of the performance advantage.

An optional second argument (an integer between 1 and 256) indicates the
increment amount for C<P> parameter selection.  The default value of 1 yields
the parameter selection described in L</is_extra_strong_lucas_pseudoprime>,
creating a pseudoprime sequence which is a superset of the latter's
pseudoprime sequence L<OEIS A217719|http://oeis.org/A217719>.
A value of 2 yields the method used by
L<Pari|http://pari.math.u-bordeaux.fr/faq.html#primetest>.

Because the C<U = 0> condition is ignored, this produces about 5% more
pseudoprimes than the extra-strong Lucas test.  However this is still only
66% of the number produced by the strong Lucas-Selfridge test.  No BPSW
counterexamples have been found with any of the Lucas tests described.


=head2 is_euler_plumb_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and passes
Colin Plumb's Euler Criterion primality test, and returns 0 otherwise.
Pseudoprimes to this test
are a subset of the base 2 Fermat and Euler tests, but a superset
of the base 2 strong pseudoprime (Miller-Rabin) test.

The main reason for this test is that it is slightly more efficient
than other probable prime tests.


=head2 is_perrin_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and
C<n> divides C<P(n)> where C<P(n)> is the Perrin number of C<n>, and returns 0 otherwise.
The Perrin sequence is defined by
C<P(n) = P(n-2) + P(n-3)> with C<P(0) = 3, P(1) = 0, P(2) = 2>.

While pseudoprimes are relatively rare (the first two are 271441 and 904631),
infinitely many exist.  They have significant overlap with the base-2
pseudoprimes and strong pseudoprimes, making the test inferior to the
Lucas or Frobenius tests for combined testing.
The pseudoprime sequence is L<OEIS A013998|http://oeis.org/A013998>.

The implementation uses modular pre-filters, Montgomery math, and the
Adams/Shanks doubling method.  This is significantly more efficient than
other known implementations.

An optional second argument C<r> indicates whether to run additional tests.
With C<r=1>, C<P(-n) = -1 mod n> is also verified,
creating the "minimal restricted" test.
With C<r=2>, the full signature is also tested using the Adams and Shanks (1982)
rules (without the quadratic form test).
With C<r=3>, the full signature is tested using the Grantham (2000) test, which
additionally does not allow pseudoprimes to be divisible by 2 or 23.
The minimal restricted pseudoprime sequence is L<OEIS A018187|http://oeis.org/A018187>.


=head2 is_catalan_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and
C<< (-1)^{(n-1)/2} * C_{(n-1)/2} >> is congruent to 2 mod C<n>,
where C<C_n> is the nth Catalan number, and returns 0 otherwise.
The nth Catalan number is equal to C<binomial(2n,n)/(n+1)>.
All odd primes satisfy this condition, and only three known composites.

The pseudoprime sequence is L<OEIS A163209|http://oeis.org/A163209>.

There is no known efficient method to perform the Catalan primality test,
so it is a curiosity rather than a practical test.  The implementation uses
a method from Charles Greathouse IV (2015) and results from
Aebi and Cairns (2008) to produce results many orders of magnitude faster
than other known implementations, but it is still vastly slower than
other compositeness tests.


=head2 is_frobenius_pseudoprime

Given an integer C<n> and two optional integer parameters C<a> and C<b>,
returns 1 if C<n> is positive and a Frobenius probable prime with respect
to the polynomial C<x^2 - ax + b>, and returns 0 otherwise.
Without the parameters, C<b = 2> and
C<a> is the least positive odd number such that C<(a^2-4b|n) = -1>.
This selection has no pseudoprimes below C<2^64> and none known.  In any
case, the discriminant C<a^2-4b> must not be a perfect square.

Some authors use the Fibonacci polynomial C<x^2-x-1> corresponding to
C<(1,-1)> as the default method for a Frobenius probable prime test.
This creates a weaker test than most other parameter choices (e.g. over
twenty times more pseudoprimes than C<(3,-5)>), so is not used as the
default here.  With the C<(1,-1)> parameters the pseudoprime sequence
is L<OEIS A212424|http://oeis.org/A212424>.

The Frobenius test is a stronger test than the Lucas test.  Any Frobenius
C<(a,b)> pseudoprime is also a Lucas C<(a,b)> pseudoprime but the converse
is not true, as any Frobenius C<(a,b)> pseudoprime is also a Fermat pseudoprime
to the base C<|b|>.  We can see that with the default parameters this is
similar to, but somewhat weaker than, the BPSW test used by this module
(which uses the strong and extra-strong versions of the probable prime and
Lucas tests respectively).

Also see the more efficient L</is_frobenius_khashin_pseudoprime> and
L</is_frobenius_underwood_pseudoprime> which have no known counterexamples
and run quite a bit faster.


=head2 is_frobenius_underwood_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and passes the
efficient Frobenius test of Paul Underwood, and returns 0 otherwise.
This selects a parameter C<a> as the least non-negative integer such that
C<(a^2-4|n)=-1>, then verifies that C<(x+2)^(n+1) = 2a + 5 mod (x^2-ax+1,n)>.
This combines a Fermat and Lucas test with a cost of only slightly more
than 2 strong pseudoprime tests.
This makes it similar to, but faster than, a regular Frobenius test.

There are no known pseudoprimes to this test and extensive computation has
shown no counterexamples under C<2^50>.  This test also has no overlap
with the BPSW test, making it a very effective method for adding additional
certainty.
Performance at 1e12 is about 60% slower than BPSW.

=head2 is_frobenius_khashin_pseudoprime

Given an integer C<n>, returns 1 if C<n> is positive and
passes the Frobenius test of Sergey Khashin, and returns 0 otherwise.
The test verifies C<n> is not a perfect square,
selects the parameter C<c> as the smallest odd prime such that C<(c|n)=-1>,
then verifies that C<(1+D)^n = (1-D) mod n> where C<D = sqrt(c) mod n>.

There are no known pseudoprimes to this test and Khashin (2018) shows
there are no counterexamples under C<2^64>.
Performance at 1e12 is about 40% slower than BPSW.

=head2 miller_rabin_random

Given an integer C<n> and a positive integer C<k>,
returns 1 if C<n> is positive and passes C<k> Miller-Rabin tests
using uniform random bases selected between C<2> and C<n-2>.

This should not be used in place of L</is_prob_prime>, L</is_prime>,
or L</is_provable_prime>.  Those functions will be faster and provide
better results than running C<k> Miller-Rabin tests.  This function can
be used if one wants more assurances for non-proven primes, such as for
cryptographic uses where the size is large enough that proven primes are
not desired.



=head2 is_prob_prime

  my $prob_prime = is_prob_prime($n);
  # Returns 0 (composite), 2 (prime), or 1 (probably prime)

Given an integer C<n>, returns 0 (composite),
2 (definitely prime), or 1 (probably prime).

For 64-bit input (native or bignum), this uses either a deterministic set of
Miller-Rabin tests (1, 2, or 3 tests) or a strong BPSW test consisting of a
single base-2 strong probable prime test followed by a strong Lucas test.
This has been verified with Jan Feitsma's 2-PSP database to produce no false
results for 64-bit inputs.  Hence the result will always be 0 (composite) or
2 (prime).

For inputs larger than C<2^64>, an extra-strong Baillie-PSW primality test is
performed (also called BPSW or BSW).  This is a probabilistic test, so only
0 (composite) and 1 (probably prime) are returned.  There is a possibility that
composites may be returned marked prime, but since the test was published in
1980, not a single BPSW pseudoprime has been found, so it is extremely likely
to be prime.
While we believe (Pomerance 1984) that an infinite number of counterexamples
exist, there is a weak conjecture (Martin) that none exist under 10000 digits.


=head2 is_bpsw_prime

Given an integer C<n>, returns 0 (composite), 2 (definitely prime),
or 1 (probably prime), using the BPSW primality test (extra-strong variant).
Normally one of the L<Math::Prime::Util/is_prime> or
L<Math::Prime::Util/is_prob_prime> functions will suffice, but those
functions do pre-tests to find easy composites.  If you know this is not
necessary, then calling L</is_bpsw_prime> may save a small amount of time.


=head2 is_provable_prime

  say "$n is definitely prime" if is_provable_prime($n) == 2;

Given an integer C<n>, returns 0 (composite), 2 (definitely prime),
or 1 (probably prime).  This gives it the same return
values as L</is_prime> and L</is_prob_prime>.  Note that numbers below 2^64
are considered proven by the deterministic set of Miller-Rabin bases or the
BPSW test.  Both of these have been tested for all small (64-bit) composites
and do not return false positives.

Using the L<Math::Prime::Util::GMP> module is B<highly recommended> for doing
primality proofs, as it is much, much faster.  The pure Perl code is just not
fast for this type of operation, nor does it have the best algorithms.
It should suffice for proofs of up to 40 digit primes, while the latest
MPU::GMP works for primes of hundreds of digits (thousands with an optional
larger polynomial set).

The pure Perl implementation uses theorem 5 of BLS75 (Brillhart, Lehmer, and
Selfridge's 1975 paper), an improvement on the Pocklington-Lehmer test.
This requires C<n-1> to be factored to C<(n/2)^(1/3))>.  This is often fast,
but as C<n> gets larger, it takes exponentially longer to find factors.

L<Math::Prime::Util::GMP> implements both the BLS75 theorem 5 test as well
as ECPP (elliptic curve primality proving).  It will typically try a quick
C<n-1> proof before using ECPP.  Certificates are available with either method.
This results in proofs of 200-digit primes in under 1 second on average, and
many hundreds of digits are possible.  This makes it significantly faster
than Pari 2.1.7's C<is_prime(n,1)> which is the default for L<Math::Pari>.


=head2 prime_certificate

  my $cert = prime_certificate($n);
  say verify_prime($cert) ? "proven prime" : "not prime";

Given an integer C<n>, returns a primality certificate
as a multi-line string.  If we could not prove C<n> prime, an empty
string is returned (C<n> may or may not be composite).
This may be examined or given to L</verify_prime> for verification.  The latter
function contains the description of the format.


=head2 is_provable_prime_with_cert

Given an integer C<n>, returns a two element array containing
the result of L</is_provable_prime>:
  0  definitely composite
  1  probably prime
  2  definitely prime
and a primality certificate like L</prime_certificate>.
The certificate will be an empty string if the first element is not 2.


=head2 verify_prime

  my $cert = prime_certificate($n);
  say verify_prime($cert) ? "proven prime" : "not prime";

Given a primality certificate, returns either 0 (not verified)
or 1 (verified).  Most computations are done using pure Perl with
Math::BigInt, so you probably want to install and use Math::BigInt::GMP,
and ECPP certificates will be faster with Math::Prime::Util::GMP for
its elliptic curve computations.

If the certificate is malformed, the routine will carp a warning in addition
to returning 0.  If the C<verbose> option is set (see L</prime_set_config>)
then if the validation fails, the reason for the failure is printed in
addition to returning 0.  If the C<verbose> option is set to 2 or higher, then
a message indicating success and the certificate type is also printed.

A certificate may have arbitrary text before the beginning (the primality
routines from this module will not have any extra text, but this way
verbose output from the prover can be safely stored in a certificate).
The certificate begins with the line:

  [MPU - Primality Certificate]

All lines in the certificate beginning with C<#> are treated as comments
and ignored, as are blank lines.  A version number may follow, such as:

  Version 1.0

For all inputs, base 10 is the default, but at any point this may be
changed with a line like:

  Base 16

where allowed bases are 10, 16, and 62.  This module will only use base 10,
so its routines will not output Base commands.

Next, we look for (using "100003" as an example):

  Proof for:
  N 100003

where the text C<Proof for:> indicates we will read an C<N> value.  Skipping
comments and blank lines, the next line should be "N " followed by the number.

After this, we read one or more blocks.  Each block is a proof of the form:

  If Q is prime, then N is prime.

Some of the blocks have more than one Q value associated with them, but most
only have one.  Each block has its own set of conditions which must be
verified, and this can be done completely self-contained.  That is, each
block is independent of the other blocks and may be processed in any order.
To be a complete proof, each block must successfully verify.  The block
types and their conditions are shown below.

Finally, when all blocks have been read and verified, we must ensure we
can construct a proof tree from the set of blocks.  The root of the tree
is the initial C<N>, and for each node (block), all C<Q> values must
either have a block using that value as its C<N> or C<Q> must be less
than C<2^64> and pass BPSW.

Some other certificate formats (e.g. Primo) use an ordered chain, where
the first block must be for the initial C<N>, a single C<Q> is given which
is the implied C<N> for the next block, and so on.  This simplifies
validation implementation somewhat, and removes some redundant
information from the certificate, but has no obvious way to add proof
types such as Lucas or the various BLS75 theorems that use multiple
factors.  I decided that the most general solution was to have the
certificate contain the set in any order, and let the verifier do the
work of constructing the tree.

The blocks begin with the text "Type ..." where ... is the type.  One or
more values follow.  The defined types are:

=over 4

=item C<Small>

  Type Small
  N 5791

N must be less than 2^64 and be prime (use BPSW or deterministic M-R).

=item C<BLS3>

  Type BLS3
  N  2297612322987260054928384863
  Q  16501461106821092981
  A  5

A simple n-1 style proof using BLS75 theorem 3.  This block verifies if:
  a  Q is odd
  b  Q > 2
  c  Q divides N-1
  .  Let M = (N-1)/Q
  d  MQ+1 = N
  e  M > 0
  f  2Q+1 > sqrt(N)
  g  A^((N-1)/2) mod N = N-1
  h  A^(M/2) mod N != N-1

=item C<Pocklington>

  Type Pocklington
  N  2297612322987260054928384863
  Q  16501461106821092981
  A  5

A simple n-1 style proof using generalized Pocklington.  This is more
restrictive than BLS3 and much more than BLS5.  This is Primo's type 1,
and this module does not currently generate these blocks.
This block verifies if:
  a  Q divides N-1
  .  Let M = (N-1)/Q
  b  M > 0
  c  M < Q
  d  MQ+1 = N
  e  A > 1
  f  A^(N-1) mod N = 1
  g  gcd(A^M - 1, N) = 1

=item C<BLS15>

  Type BLS15
  N  8087094497428743437627091507362881
  Q  175806402118016161687545467551367
  LP 1
  LQ 22

A simple n+1 style proof using BLS75 theorem 15.  This block verifies if:
  a  Q is odd
  b  Q > 2
  c  Q divides N+1
  .  Let M = (N+1)/Q
  d  MQ-1 = N
  e  M > 0
  f  2Q-1 > sqrt(N)
  .  Let D = LP*LP - 4*LQ
  g  D != 0
  h  Jacobi(D,N) = -1
  .  Note: V_{k} indicates the Lucas V sequence with LP,LQ
  i  V_{m/2} mod N != 0
  j  V_{(N+1)/2} mod N == 0

=item C<BLS5>

  Type BLS5
  N  8087094497428743437627091507362881
  Q[1]  98277749
  Q[2]  3631
  A[0]  11
  ----

A more sophisticated n-1 proof using BLS theorem 5.  This requires N-1 to
be factored only to C<(N/2)^(1/3)>.  While this looks much more complicated,
it really isn't much more work.  The biggest drawback is just that we have
multiple Q values to chain rather than a single one.  This block verifies if:

  a  N > 2
  b  N is odd
  .  Note: the block terminates on the first line starting with a C<->.
  .  Let Q[0] = 2
  .  Let A[i] = 2 if Q[i] exists and A[i] does not
  c  For each i (0 .. maxi):
  c1   Q[i] > 1
  c2   Q[i] < N-1
  c3   A[i] > 1
  c4   A[i] < N
  c5   Q[i] divides N-1
  . Let F = N-1 divided by each Q[i] as many times as evenly possible
  . Let R = (N-1)/F
  d  F is even
  e  gcd(F, R) = 1
  . Let s = integer    part of R / 2F
  . Let f = fractional part of R / 2F
  . Let P = (F+1) * (2*F*F + (r-1)*F + 1)
  f  n < P
  g  s = 0  OR  r^2-8s is not a perfect square
  h  For each i (0 .. maxi):
  h1   A[i]^(N-1) mod N = 1
  h2   gcd(A[i]^((N-1)/Q[i])-1, N) = 1

=item C<ECPP>

  Type ECPP
  N  175806402118016161687545467551367
  A  96642115784172626892568853507766
  B  111378324928567743759166231879523
  M  175806402118016177622955224562171
  Q  2297612322987260054928384863
  X  3273750212
  Y  82061726986387565872737368000504

An elliptic curve primality block, typically generated with an Atkin/Morain
ECPP implementation, but this should be adequate for anything using the
Atkin-Goldwasser-Kilian-Morain style certificates.
Some basic elliptic curve math is needed for these.
This block verifies if:

  .  Note: A and B are allowed to be negative, with -1 not uncommon.
  .  Let A = A % N
  .  Let B = B % N
  a  N > 0
  b  gcd(N, 6) = 1
  c  gcd(4*A^3 + 27*B^2, N) = 1
  d  Y^2 mod N = X^3 + A*X + B mod N
  e  M >= N - 2*sqrt(N) + 1
  f  M <= N + 2*sqrt(N) + 1
  g  Q > (N^(1/4)+1)^2
  h  Q < N
  i  M != Q
  j  Q divides M
  .  Note: EC(A,B,N,X,Y) is the point (X,Y) on Y^2 = X^3 + A*X + B, mod N
  .        All values work in affine coordinates, but in theory other
  .        representations work just as well.
  .  Let POINT1 = (M/Q) * EC(A,B,N,X,Y)
  .  Let POINT2 = M * EC(A,B,N,X,Y)  [ = Q * POINT1 ]
  k  POINT1 is not the identity
  l  POINT2 is the identity

=back

=head2 is_aks_prime

  say "$n is definitely prime" if is_aks_prime($n);

Given an integer C<n>, returns 1 if C<n> is positive and
passes the Agrawal-Kayal-Saxena (AKS) primality test, and returns 0 otherwise.
This is a deterministic unconditional primality test which runs
in polynomial time for general input.

While this is an important theoretical algorithm, and makes an interesting
example, it is hard to overstate just how impractically slow it is in
practice.  It is not used for any purpose in non-theoretical work, as it is
literally B<millions> of times slower than other algorithms.  From R.P.
Brent, 2010:  "AKS is not a practical algorithm.  ECPP is much faster."
This module also has ECPP, and indeed it is much faster.

This implementation uses theorem 4.1 from Bernstein (2003).  It runs
substantially faster than the original, v6 revised paper with Lenstra
improvements, or the late 2002 improvements of Voloch and Bornemann.
The GMP implementation uses a binary segmentation method for modular
polynomial multiplication (see Bernstein's 2007 Quartic paper), which
reduces to a single scalar multiplication, at which GMP excels.
Because of this, the GMP implementation is likely to be faster once
the input is larger than C<2^33>.


=head2 is_mersenne_prime

  say "2^607-1 (M607) is a Mersenne prime" if is_mersenne_prime(607);

Given an integer C<p>, returns 1 if C<p> is positive and
the Mersenne number C<2^p-1> is prime, and returns 0 otherwise.
Since an enormous effort has gone into testing these, a list of known
Mersenne primes is used to accelerate this.  Beyond the highest sequential
Mersenne prime (currently 37,156,667) this performs pretesting followed by
the Lucas-Lehmer test.

The Lucas-Lehmer test is a deterministic unconditional test that runs
very fast compared to other primality methods for numbers of comparable
size, and vastly faster than any known general-form primality proof methods.
While this test is fast, the GMP implementation is not nearly as fast as
specialized programs such as C<prime95>.  Additionally, since we use the
table for "small" numbers, testing via this function call will only occur
for numbers with over 9.8 million digits.  At this size, tools such as
C<prime95> are greatly preferred.


=head2 is_ramanujan_prime

Given an integer C<n>, returns 1 if C<n> is positive and
is a Ramanujan prime, and returns 0 otherwise.
Therefore, numbers that can be produced
by the functions L</ramanujan_primes> and L</nth_ramanujan_prime> will
return 1, while all other numbers will return 0.

There is no simple function for this predicate, so Ramanujan primes through
at least C<n> are generated, then a search is performed for C<n>.  This is
not efficient for multiple calls.


=head2 is_gaussian_prime

  say is_gaussian_prime(3,0);  # "2"  :  3   => 3 mod 4 => prime
  say is_gaussian_prime(1,1);  # "2"  :  1+i => norm 2  => prime
  say is_gaussian_prime(5,0);  # "0"  :  5   => 1 mod 4 => (2+i)(2-i)

Given two integers C<a> and C<b>, returns either 0, 1, or 2 to indicate
whether C<n = a+bi> is, respectively, a Gaussian composite,
probable Gaussian prime, or definite Gaussian prime.
This is true if and only if one of:

=over 4

=item C<a = 0> and |b| is a prime congruent to 3 modulo 4.

=item C<b = 0> and |a| is a prime congruent to 3 modulo 4.

=item C<a> and C<b> are nonzero and C<a^2 + b^2> is prime.

=back


=head2 is_delicate_prime

Given an integer C<n>, returns 1 if C<n> is positive and
is a digitally delicate prime, and returns 0 otherwise.
These are numbers which are prime, but changing any single base-10 digit
always produces a composite number.

An optional second argument is the base C<base> which must be at least 2.
This is the base used for changing digits to check for compositeness.

These are variously called "weakly prime" or "digitally delicate prime"
numbers.
Note that the first digit can be changed to a zero.

Variations not considered here include
making changing the first digit restricted to non-zero (OEIS A158124)
and allowing leading zero digits to be changed ("widely DDPs").

This is the L<OEIS series A050249|http://oeis.org/A050249>.
With different bases, this is L<OEIS series A186995|http://oeis.org/A186995>.


=head2 is_odd

Given an integer C<n>, returns 1 if C<n> is odd and 0 otherwise.

=head2 is_even

Given an integer C<n>, returns 1 if C<n> is even and 0 otherwise.

=head2 is_divisible

Given integers C<n> and C<d>, returns 1 if C<|n|> is exactly divisible
by C<|d|>, and 0 otherwise.

This corresponds to the GMP function C<mpz_divisible_p>.
This includes its semantics with C<d=0> which returns 0 unless C<n=0>.

More than one divisor can be given, e.g. C<is_divisible(1001,2,3,5,7)>,
allowing one to test multiple divisors with one call.  The result is 1
if C<n> is exactly divisible by any of the C<d> values, and 0 if it
is divisible by none of them.

=head2 is_congruent

Given integers C<n>, C<c>, and C<d>, returns 1 if C<n> is congruent to C<c>
modulo C<|d|>, and 0 otherwise.

This corresponds to the GMP function C<mpz_congruent_p>.
This includes its semantics with C<d=0> which returns 0 unless C<n=c>.

=head2 is_perfect_number

Given integer C<n>, returns 1 if C<n> is a positive integer that is the
sum of its divisors excluding the number itself, or equivalently a number
that is equal to its aliquot sum.

=head2 is_power

  say "$n is a perfect square" if is_power($n, 2);
  say "$n is a perfect cube" if is_power($n, 3);
  say "$n is a ", is_power($n), "-th power";

Given a single integer input C<n>, returns k if C<n = r^k> for
some integer C<< r > 1, k > 1 >>, and 0 otherwise.  The k returned is
the largest possible.  This can be used in a boolean statement to
determine if C<n> is a perfect power.

If given an integer C<n> and a non-negative integer C<k>,
returns 1 if C<n> is a C<k-th> power, and 0 otherwise.
For example, if C<k=2> then this detects perfect squares.
Setting C<k=0> gives behavior like the first case (the largest root is found
and its value is returned).

If a third argument is given, it must be a scalar reference.  If C<n> is
a k-th power, then this will be set to the k-th root of C<n>.  For example:

  my $n = 222657534574035968;
  if (my $pow = is_power($n, 0, \my $root)) { say "$n = $root^$pow" }
  # prints:  222657534574035968 = 2948^5

This corresponds to Pari/GP's C<ispower> function with integer arguments.


=head2 is_square

Given an integer C<n>, returns 1 if C<n> is a perfect square,
and returns 0 otherwise.  This is identical to C<is_power(n,2)>.

This corresponds to Pari/GP's C<issquare> function.

=head2 is_sum_of_squares

Given an integer C<n> and an optional positive integer number of squares C<k>,
returns 1 if C<|n|> can be represented as the sum of exactly C<k> squares.
C<k> defaults to 2.
All positive integers can be represented by 4 or more squares, so
only C<k == 2> and C<k == 3> are interesting cases.

With C<k == 2> this produces the sequence
L<OEIS A001481|http://oeis.org/A001481>.
With C<k == 3> this produces the sequence
L<OEIS A000378|http://oeis.org/A000378>.

=head2 is_powerfree

Given an integer C<n> and an optional non-negative integer C<k>, returns
1 if C<|n|> has no divisor C<d^k>, and returns 0 otherwise.
This determines if C<|n|> has any k-th (or higher) powers in the prime
factorization.
C<k> defaults to 2.

With C<k == 2> this produces the sequence of square-free integers
L<OEIS A005117|http://oeis.org/A005117>.
With C<k == 3> this produces the sequence of cube-free integers
L<OEIS A004709|http://oeis.org/A004709>.
With C<k == 4> this produces the sequence of biquadrate-free integers
L<OEIS A046100|http://oeis.org/A046100>.

=head2 powerfree_count

Given an integer C<n> and an optional non-negative integer C<k>, returns
the number of k-powerfree positive integers less than or equal to C<n>.
C<k> defaults to 2.

With C<k == 2> this produces the sequence
L<OEIS A013928|http://oeis.org/A013928>.
With C<k == 3> this produces the sequence
L<OEIS A060431|http://oeis.org/A060431>.

=head2 nth_powerfree

Given a non-negative integer C<n> and an optional non-negative integer C<k>,
returns the C<n>-th k-powerfree number.
If C<k> is omitted, C<k=2> is used.
Returns undef if C<k> is less than 2 or C<n=0>.  Returns 1 for C<n=1>.

With C<k == 2> this produces the sequence
L<OEIS A005117|http://oeis.org/A005117>.
With C<k == 3> this produces the sequence
L<OEIS A004709|http://oeis.org/A004709>.

=head2 powerfree_sum

Given an integer C<n> and an optional non-negative integer C<k>, returns
the sum of k-powerfree positive integers less than or equal to C<n>.
C<k> defaults to 2.

With C<k == 2> this produces the sequence
L<OEIS A066779|http://oeis.org/A066779>.

=head2 powerfree_part

Given an integer C<n> and an optional non-negative integer C<k>, returns
the k-powerfree part of C<n>.  This is done via removing "excess" powers,
i.e. in the prime factorization of C<n>, we reduce any exponents C<E>
from C<P^E> to C<P^(E % k)>.  Alternately we can say all k-th powers are
divided out.
C<k> defaults to 2.

When C<k == 2>, this is also sometimes called C<core(n)>.  It is the
unique square-free integer C<d> such that C<n/d> is a square.

With C<k == 2> this produces the sequence
L<OEIS A007913|http://oeis.org/A007913>.
With C<k == 3> this produces the sequence
L<OEIS A050985|http://oeis.org/A050985>.

With C<k == 2> (the default), this corresponds to Pari/GP's C<core>
function and Sage's C<squarefree_part> function.

=head2 powerfree_part_sum

Given an integer C<n> and an optional non-negative integer C<k>, returns
the sum of k-powerfree parts of all positive integers C<< <= n >>.  This
is equivalent to

    vecsum(map { powerfree_part($_,$k) } 1..$n)

but substantially faster.

With C<k == 2> this produces the sequence
L<OEIS A069891|http://oeis.org/A069891>.

=head2 squarefree_kernel

Given an integer C<n>, returns the square-free kernel of C<n>.  This is
also known as the integer radical.  It is the largest square-free divisor
of C<n>, which is also the product of the distinct primes dividing C<n>.

We choose to accept negative inputs, with the result matching the input sign.

This is the L<OEIS series A007947|http://oeis.org/A007947>.

=head2 sqrtint

Given a non-negative integer input C<n>, returns the integer square root.
For native integers, this is equal to C<int(sqrt(n))>.

This corresponds to Pari/GP's C<sqrtint> function.


=head2 rootint

Given a non-negative integer C<n> and positive exponent C<k>, return the
integer k-th root of C<n>.  This is the largest integer C<r> such that
C<< r^k <= n >>.

If a third argument is present, it must be a scalar reference.
It will be set to C<r^k>.

Technically if C<n> is negative and C<k> is odd, the root exists and is
equal to C<sign(n) * |rootint(abs(n),k)>.  It was decided to follow the
behavior of Pari/GP and Math::BigInt and disallow negative C<n>.

This corresponds to Pari/GP's C<sqrtnint> function.


=head2 logint

  say "decimal digits: ", 1+logint($n, 10);
  say "digits in base 12: ", 1+logint($n, 12);
  my $be; my $e = logint(1000, 2, \$be);
  say "largest power of 2 less than or equal to 1000:  2^$e = $be";

Given a non-zero positive integer C<n> and an integer base C<b> greater
than 1, returns the largest integer C<e> such that C<< b^e <= n >>.

If a third argument is present, it must be a scalar reference.
It will be set to C<b^e>.

This corresponds to Pari/GP's C<logint> function.


=head2 lshiftint

Given an integer C<n> and an optional integer number of bits C<k>,
perform a left shift of C<n> by C<k> bits.
If the second argument is not provided, it is assumed to be 1.
This is equivalent to multiplying by C<2^k>.

With negative C<n>, this behaves as described above.  This is similar to
how Perl behaves with C<use integer> or C<use bigint>, but raw Perl
coerces the argument into an unsigned before left shifting, which is
unlikely to ever be what is wanted.

If C<k> is negative, a right shift is performed by C<|k|> bits.

This corresponds to Pari/GP's C<shift> function with a positive number
of bits, and Mathematica's C<BitShiftLeft> function.

=head2 rshiftint

Given an integer C<n> and an optional integer number of bits C<k>,
perform a right shift of C<n> by C<k> bits.
If the second argument is not provided, it is assumed to be 1.
This is equivalent to truncated division by C<2^k>.

With a negative C<n>, the result is equal to C<-rshiftint(-n,k)>.
This means it is not "arithmetic right shift" or "logical right shift"
as commonly used with fixed-width registers in a particular bit format,
but instead treated as sign and magnitude, where the magnitude
is right shifted.

If C<k> is negative, a left shift is performed by C<|k|> bits.

For an interesting discussion of arithmetic right shift, see
Guy Steele's 1977 article "Arithmetic Shift Considered Harmful".

This corresponds to Pari/GP's C<shift> function with a negative number
of bits, and Mathematica's C<BitShiftRight> function.  The result is equal
to dividing by the power of 2 using L</tdivrem> or GMP's C<mpz_tdiv_q_2exp>.

=head2 rashiftint

Given an integer C<n> and an optional integer number of bits C<k>,
perform a signed arithmetic right shift of C<n> by C<k> bits.
If the second argument is not provided, it is assumed to be 1.
This is equivalent to floor division by C<2^k>.

If C<k> is negative, a left shift is performed by C<|k|> bits.

For non-negative C<n>, this is always equal to L</rshiftint>.
With negative arguments it is similar to L<Math::BigInt#brsft>, Python,
and Java's BigInteger, which use floor division by C<2^k>.  The result is equal
to dividing by the power of 2 using L</divint> or GMP's C<mpz_fdiv_q_2exp>.


=head2 signint

Given an integer C<n>, returns the sign of C<n>.
Returns -1, 0, or 1 if C<n> is negative, zero, or positive respectively.

This corresponds to Pari/GP's C<sign> function, GMP's C<mpz_sgn> function,
Raku's C<sign> method, and Math::BigInt's C<sign> method.
Some of those extend to non-integers.

=head2 cmpint

Given integers C<a> and C<b>, returns -1, 0, or 1 if C<a> is respectively
less than, equal to, or greater than C<b>.

The main value of this is to ensure Perl never silently converts the values
to floating point, which can give wrong results, and also avoid having to
manually convert everything to bigints.

This corresponds to Pari/GP's C<cmp> function, GMP's C<mpz_cmp> function,
Math::BigInt's C<bcmp> method, and Perl's << <=> >> operator.
Prior to version 6.2, GMP could return negative or positive values other
than -1 and 1.

=head2 addint

Given integers C<a> and C<b>, returns C<a + b>.

These integer arithmetic functions (C<addint>, C<subint>, C<mulint>,
C<add1int>, C<sub1int>, C<absint>, C<negint>) exist to offer exact
integer arithmetic without overflow or NV conversion, while returning
native integers when they fit, and bigints only when needed.
Other choices include:

=over 4

=item *

B<Perl native operations.>  This is fine with small numbers, but once
large enough, values will be converted to floating point (NV).  This
means incorrect results.  Values larger than 64-bit are completely
unsupported.  One might expect C<2^53> to be the usual point for
"large enough", but not only is the NV type platform dependent, but
very old 64-bit Perl will aggressively convert values to NV starting
at C<2^49> even with NV being a IEEE-754 double.

=item *

B<use integer.>  Gives exact integer math as if we were using C<IV> types
in C.  We are still left with 32-bit versus 64-bit platform differences,
being restricted to signed type, and no support for larger values.

=item *

B<Math::BigInt, Math::GMPz, etc.>  If one knows large values will be
used, this is a good idea.  Use bigint objects for all values, and all
operations are methods on the objects and give correct results.
This is functionally a good solution, but it will be 10 to B<500>
times slower as well as more memory.

=back

All these functions accept native integers (IV/UV), bigints, and string
representations of integers.  Results will be in native types if possible,
and as objects of the chosen bigint class otherwise.  Best performance
will still be had by native operations within range, or by using fast
classes like L<Math::GMPz> if most operations need it.
We give correct behavior while only paying the performance penalty when
needed, although there is still some overhead since we are not built
into the language like Raku or Python.

=head2 subint

Given integers C<a> and C<b>, returns C<a - b>.

=head2 add1int

Given integer C<n>, returns C<n + 1>.

=head2 sub1int

Given integer C<n>, returns C<n - 1>.

=head2 mulint

Given integers C<a> and C<b>, returns C<a * b>.

=head2 powint

Given an integer C<a> and a non-negative integer C<b>,
returns C<a^b>.  C<0^0> will return 1.

The exponent C<b> is converted into an unsigned long.

=head2 divint

Given integers C<a> and C<b>, returns the quotient C<a / b>.

Floor division is used, so q is rounded towards C<-inf> and
the remainder has the same sign as the divisor C<b>.
This is the same as modern L<Math::BigInt/bdiv>,
GMP C<fdiv> functions, and Python's integer division.

For negative inputs, this will not be identical to native Perl division,
which oddly uses a truncated quotient and floored remainder.
More importantly, consistent and correct 64-bit integer division in
Perl is problematic.

Pari/GP's C<\\> integer division operator uses Euclidean division,
which matches their C<divrem> function.  Our C<divint> and C<modint>
operators both use floor division, which matches Raku and Python.
We also have Euclidean, truncated, and ceiling division available via
L</divrem>, L</tdivrem>, and L</cdivrem> respectively.

=head2 modint

Given integers C<a> and C<b>, returns the modulo C<a % b>.

    r = a - b * floor(a / b)

Floor division is used, so q is rounded towards C<-inf>
and r has the same sign as the divisor C<b>.
This is the same as modern L<Math::BigInt/bmod> and the
GMP C<fdiv> functions.

Like with C<divint>, we use floor division, while Pari/GP uses Euclidean
for their C<%> integer remainder operator.

=head2 cdivint

Given integers C<a> and C<b>, returns the quotient C<a / b>.

Ceiling division is used, so q is rounded towards C<+inf> and
the remainder has the opposite sign as the divisor C<b>.

=head2 divrem

    my($quo, $rem) = divrem($a, $b);

Given integers C<a> and C<b>, returns a list of two items:
the Euclidean quotient and the Euclidean remainder.
The remainder is always non-negative (C<< 0 <= r < |b| >>), and the
quotient is chosen to satisfy C<< a = b*q + r >>.

This corresponds to Pari/GP's C<divrem> function.
There is no explicit function in L<Math::BigInt> that gives
this division method for signed inputs.

=head2 tdivrem

Given integers C<a> and C<b>, returns a list of two items:
the truncated quotient and the truncated remainder.

The resulting pair will match
L<Math::BigInt/btdiv> and L<Math::BigInt/btmod>.
This matches C99 "truncation toward zero" semantics as well.

=head2 fdivrem

Given integers C<a> and C<b>, returns a list of two items:
the floored quotient and the floored remainder.
The results will match the individual L</divint> and L</modint>
functions, since they also use floored division.

This corresponds to Python's builtin C<divmod> function, and
Raku's builtin C<div> and C<mod> functions.
The resulting pair will match
L<Math::BigInt/bdiv> and L<Math::BigInt/bmod>.

=head2 cdivrem

Given integers C<a> and C<b>, returns a list of two items:
the ceiling quotient (rounded towards C<+inf>) and the ceiling remainder.
The remainder has the opposite sign from the divisor C<b>.
This allows one to perform division with rounding up.

=head2 absint

Given integer C<n>, return C<|n|>, i.e. the absolute value of C<n>.

=head2 negint

Given integer C<n>, return C<-n>.


=head2 lucasu

  say "Fibonacci($_) = ", lucasu(1,-1,$_) for 0..100;

Given integers C<P>, C<Q>, and the non-negative integer C<k>,
computes C<U_k> for the Lucas sequence defined by C<P>,C<Q>.  These include
the Fibonacci numbers (C<1,-1>), the Pell numbers (C<2,-1>), the Jacobsthal
numbers (C<1,-2>), the Mersenne numbers (C<3,2>), and more.

Also see L</lucasumod> for fast computation mod n.

This corresponds to OpenPFGW's C<lucasU> function and gmpy2's C<lucasu>
function.

=head2 lucasv

  say "Lucas($_) = ", lucasv(1,-1,$_) for 0..100;

Given integers C<P>, C<Q>, and the non-negative integer C<k>,
computes C<V_k> for the Lucas sequence defined by C<P>,C<Q>.  These include
the Lucas numbers (C<1,-1>).

Also see L</lucasvmod> for fast computation mod n.

This corresponds to OpenPFGW's C<lucasV> function and gmpy2's C<lucasv>
function.

=head2 lucasuv

  ($U, $V) = lucasuv(1,-2,17); # 17-th Jacobsthal, Jacobsthal-Lucas.

Given integers C<P>, C<Q>, and the non-negative integer C<k>,
computes both C<U_k> and C<V_k> for the Lucas sequence defined
by C<P>,C<Q>.
Generating both values is typically not much more time than one.

Also see L</lucasuvmod> for fast computation mod n.

=head2 gcd

Given a list of integers, returns the greatest common divisor.  This is
often used to test for L<coprimality|https://oeis.org/wiki/Coprimality>.

Each input C<n> is treated as C<|n|>.

=head2 lcm

Given a list of integers, returns the least common multiple.  Note that we
follow the semantics of Mathematica, Pari, and Raku, re:

  lcm(0, n) = 0              Any zero in list results in zero return
  lcm(n,-m) = lcm(n, m)      We use the absolute values
  lcm() = 1                  lcm of empty list returns 1

=head2 gcdext

Given two integers C<x> and C<y>, returns C<u,v,d> such that C<d = gcd(x,y)>
and C<u*x + v*y = d>.  This uses the extended Euclidean algorithm to compute
the values satisfying Bézout's Identity.

This corresponds to Pari's C<gcdext> function, which was renamed from
C<bezout> in Pari 2.6.  The results will hence match L<Math::Pari/bezout>.

=head2 chinese

  say chinese( [14,643], [254,419], [87,733] );  # 87041638

Solves a system of simultaneous congruences using the Chinese Remainder
Theorem (with extension to non-coprime moduli).  A list of C<[a,n]> pairs
are taken as input, each representing an equation C<x ≡ a mod |n|>.  If no
solution exists, C<undef> is returned.  If a solution is returned, the
modulus is equal to the lcm of all the given moduli (see L</lcm>).  In
the standard case where all values of C<n> are coprime, this is just the
product.
The C<a> values must be integers, while the C<n> values must be
non-zero integers.  Like other mod functions, we use C<abs(n)>.

Comparison to similar functions in other software:

  Math::ModInt::ChineseRemainder:
    cr_combine( mod(a1,m1), mod(a2,m2), ... )

  Pari/GP:
    chinese( [Mod(a1,m1), Mod(a2,m2), ...] )

  Mathematica:
    ChineseRemainder[{a1, a2, ...}, {m1, m2, ...}]

  SAGE:
    crt( [a1,m1], [a2,m2], ... )
    crt(a1,m1,a2,m2,...)
    CRT_list( [a1,a2,...], [m1,m2,...] )

=head2 chinese2

Functions like L</chinese> but returns two items: the remainder
and the modulus.
If a solution exists, the second value (the final modulus) is equal to
the lcm of the absolute values of all the given moduli.

If no solution exists, both return values will be C<undef>.

=head2 frobenius_number

Finds the Frobenius number of a set of positive integers.
This is the largest positive integer that cannot be represented
as a non-negative linear combination of the input set.  Each set element
must be positive (all elements greater than zero) and setwise coprime:
C<gcd(a1,a2,...,an) = 1>.

This is sometimes called the "coin problem".

This corresponds to Mathematica's C<FrobeniusNumber> function.  Matching
their API, we return -1 if any set element is C<1>.

=head2 vecsum

  say "Totient sum 500,000: ", vecsum(euler_phi(0,500_000));

Returns the sum of all arguments, each of which must be an integer.  This
is similar to List::Util's L<List::Util/sum0> function, but has a very
important difference.  List::Util turns all inputs into doubles and returns
a double, which will mean incorrect results with large integers.  C<vecsum>
sums (signed) integers and returns the untruncated result.

Processing is done on native integers while possible, including using a
128-bit running sum in the C code.

=head2 vecprod

  say "Totient product 5,000: ", vecprod(euler_phi(1,5_000));

Returns the product of all arguments, each of which must be an integer.  This
is similar to List::Util's L<List::Util/product> function, but keeps all
results as integers and automatically switches to bigints if needed.

=head2 vecmin

  say "Smallest Totient 100k-200k: ", vecmin(euler_phi(100_000,200_000));

Returns the minimum of all arguments, each of which must be an integer.
This is similar to List::Util's L<List::Util/min> function, but has a very
important difference.  List::Util turns all inputs into doubles and returns
a double, which gives incorrect results with large integers.  C<vecmin>
validates and compares all results as integers.  The validation step will
make it a little slower than L<List::Util/min> but this prevents accidental
and unintentional use of floats.

=head2 vecmax

  say "Largest Totient 100k-200k: ", vecmax(euler_phi(100_000,200_000));

Returns the maximum of all arguments, each of which must be an integer.
This is similar to List::Util's L<List::Util/max> function, but has a very
important difference.  List::Util turns all inputs into doubles and returns
a double, which gives incorrect results with large integers.  C<vecmax>
validates and compares all results as integers.  The validation step will
make it a little slower than L<List::Util/max> but this prevents accidental
and unintentional use of floats.

=head2 vecreduce

  say "Count of non-zero elements: ", vecreduce { $a + !!$b } (0,@v);
  my $checksum = vecreduce { $a ^ $b } @{twin_primes(1000000)};

Does a reduce operation via left fold.  Takes a block and a list as arguments.
The block uses the special local variables C<a> and C<b> representing the
accumulation and next element respectively, with the result of the block being
used for the new accumulation.  No initial element is used, so C<undef>
will be returned with an empty list.

The interface is exactly the same as L<List::Util/reduce>.  This was done to
increase portability and minimize confusion.  See chapter 7 of Higher Order
Perl (or many other references) for a discussion of reduce with empty or
singular-element lists.  It is often a good idea to give an identity element
as the first list argument.

While operations like L</vecmin>, L</vecmax>, L</vecsum>, L</vecprod>, etc.
can be fairly easily done with this function, it will not be as efficient.
There are a wide variety of other functions that can be easily made with
reduce, making it a useful tool.

=head2 vecany

Returns true if any element of a list satisfies a block.  See L</vecfirst>.

=head2 vecall

Returns true if all elements of a list satisfy a block.  See L</vecfirst>.

=head2 vecnone

Returns true if no element of a list satisfies a block.  See L</vecfirst>.

=head2 vecnotall

Returns true if not all elements of a list satisfy a block.  See L</vecfirst>.

=head2 vecfirst

  say "all values are Carmichael" if vecall { is_carmichael($_) } @n;

Short circuit evaluations of a block over a list.  Takes a block and a list
as arguments.  The block is called with C<$_> set to each list element, and
evaluation on list elements is done until either all list values have been
evaluated or the result condition can be determined.  For instance, in the
example of C<vecall> above, evaluation stops as soon as any value returns
false.

The interface is exactly the same as the C<any>, C<all>, C<none>, C<notall>,
and C<first> functions in L<List::Util>.  This was done to increase
portability and minimize confusion.  Unlike other vector functions like
C<vecmin>, C<vecmax>, C<vecsum>, etc. there is no added value to using
these versus the ones from L<List::Util>.  They are here for convenience.

These operations can fairly easily be mapped to C<scalar(grep {...} @n)>,
but that does not short-circuit and is less obvious.

=head2 vecfirstidx

  say "first Carmichael is index ", vecfirstidx { is_carmichael($_) } @n;

Returns the index of the first element in a list that evaluates to true.
Just like vecfirst, but returns the index instead of the value.  Returns
-1 if the item could not be found.

This interface matches C<firstidx> and C<first_index> from L<List::MoreUtils>.

=head2 vecextract

  say "Power set: ", join(" ",vecextract(\@v,$_)) for 0..2**scalar(@v)-1;
  @word = vecextract(["a".."z"], [15, 17, 8, 12, 4]);

Extracts elements from an array reference based on a mask, with the
result returned as an array.  The mask is either an unsigned integer
which is treated as a bit mask, or an array reference containing integer
indices.

If the second argument is an integer, each bit set in the mask results in the
corresponding element from the array reference to be returned.  Bits are
read from the right, so a mask of C<1> returns the first element, while C<5>
will return the first and third.  The mask may be a bigint.

If the second argument is an array reference, then its elements will be used
as zero-based indices into the first array.  Duplicate values are allowed and
the ordering is preserved.  Given that Perl has fully functional array slices
in the language, this is for completeness with Pari/GP.
These are equivalent:

    vecextract($aref, $iref);
    @$aref[@$iref];

=head2 vecuniq

  my @vec = vecuniq(1,2,3,2,-10,-100,1);  # returns (1,2,3,-10,-100)

Given an array of integers, returns an array with all duplicate entries
removed.  The original ordering is preserved.  All values B<must> be defined.

This is similar to L<List::Util::uniqint> (the integer comparison version
of L<List::Util::uniq>).
Unlike the more generic L<List::Util::uniq> and L<List::MoreUtils::XS::uniq>,
all inputs must be integers.
With native integers, our function is 2-10x faster.

=head2 vecfreq

  # Produce frequency hash:
  my %h = vecfreq(1,2,2,2,3,1,4);   #  (1=>2, 2=>3, 3=>1, 4=>1)
  # Print most common value:
  say vecreduce { $h{$a} > $h{$b} ? $a : $b } keys %h;

Given an array of items, returns a hash with each key containing the unique
items, with the associated value being the occurrence count in the array.

This is identical to L<List::MoreUtils::frequency>.  It is typically faster
when given only native integers.

This is very similar to the Pari/GP function C<matreduce> for vectors,
and to Python's C<Counter>.

=head2 vecsingleton

  my @solo = vecsingleton(1,4,17,1,17,-8);  # (4,-8)
  # Same but slower:
  my %h = vecfreq(@n);
  my @onlyuniqs = grep { $h{$_} == 1 } @n;

Given an array of items, returns an array with all entries removed that
appear more than once in the list.  The original ordering is preserved.

This is identical to L<List::MoreUtils::singleton>.  When given only native
integers, it is typically 2 to 10x faster.

=head2 vecsort

  my @sorted = vecsort(1,2,3,2,-10,-100,1);   # returns (-100,-10,1,1,2,2,3)
  my @sorted = vecsort([1,2,3,2,-10,-100,1]); # same

Numerically (ascending) sort a list of integers.  The input is either a list
or a single array reference which holds the list.

All values must be defined and integers.
They may be any mix of native IV, native UV, strings, bigints.

Perl's built-in numerical sort can sometimes give incorrect results for
typical cases we encounter.  Prior to version 5.26 (2017), large 64-bit
integers were turned into NV (floating point) types.  With all current
versions of Perl, strings are turned into NV types even if they are the
text of a 64-bit integer.

In scalar context, C<vecsort> returns the number of items without sorting
(but after input validation).  This should be expected and what we typically
want.  E.g. if we only want the number of divisors, we call in scalar
context and get the number without requiring actual sorting.
Having the same results from
C<$x = vecsort(5,6,7)> and C<@v = vecsort(5,6,7); $x=@v;>
is what we want.
This contrasts with Perl's built-in C<sort> which in scalar context has
B<undefined> behaviour (in all current versions of C<perl> it returns undef).
In particular this forces all programs to use a workaround if they want to
return the results of sorting an array.
See L<Perl 5 issue 12803|https://github.com/Perl/perl5/issues/12803> for
some discussion with no resolution.

Using an array reference as input is slightly faster.

This is almost always faster than Perl's built-in numerical sort:
C<< @a = sort { $a <=> $b } @a >>.
See the performance section for more information.

=head2 vecsorti

  my @arr = map { irand } 1..100000;
  vecsorti \@arr;

Given an array reference of integers,
numerically (ascending) sorts the integers in-place.
The array reference is also returned for convenience.

This is more efficient than L</vecsort>.  Perl's C<sort> has this
optimization built-in when doing straightforward sorting on non-references.

=head2 vecequal

  my $is_equal = vecequal( [1,2,-3,[4,5,undef]], [1,2,-3,[4,5,undef]] );

Compare two arrays for equality, including nested arrays.  The values inside
the two input array references must be either an array reference, a scalar,
or undef.  Simple integers are tested with integer comparison, while other
scalars use string comparison.

This is a vector comparison, not set comparison, so ordering is important.
For the sake of wider applicability, non-integers are allowed.  Types other
than integers and strings (e.g. floating point values) are not guaranteed
to have consistent results.

No circular reference detection is performed.

Performance with XS is 3x to 100x faster than perl looping or modules like
Array::Compare, Data::Cmp, match::smart, List::Compare, and Algorithm::Diff.
Those modules have additional functionality so this is not a complete
comparison.


=head2 vecmex

  my $minimum_excluded = vecmex(0,1,2,4,6);  # returns 3

Given a list of non-negative integers, returns the smallest non-negative
integer that is not in the list.  C<mex> is short for "minimum excluded".
The list can be seen as a set, and the return value is the minimum
of the set complement.  Repeated values are allowed in the list.

C<vecmex>() = 0.
C<vecmex>(0,1,2,...,I<w>) = I<w>+1.


=head2 vecpmex

  my $minimum_excluded = vecpmex(1,2,4,6);  # returns 3

Given a list of positive integers, returns the smallest positive
integer that is not in the list.  C<mex> is short for "minimum excluded".
The list can be seen as a set, and the return value is the minimum
of the set complement.  Repeated values are allowed in the list.

C<vecpmex>() = 1.
C<vecpmex>(1,2,...,I<w>) = I<w>+1.


=head2 vecslide

  @pairsum = vecslide {$a+$b} 1..5;  # returns (1+2,2+3,3+4,4+5)

  say for vecslide { "$a->[0] $b->[1]" }
      (["hello","world"], ["goodbye","friends"], ["love","hate"]);
  # hello friends
  # goodbye hate


Given a code block and a list, calls the code block for each pair
in the list, setting the local C<$a> and C<$b> to the values in
each pair.

There is no restriction of what the list contains, as seen in the
second example.

This is identical to L<List::MoreUtils::slide>.


=head2 toset

   my $set = toset(52,-6,14,-6,0);  # $set = [-6,0,14,52]
   say "number of elements in set: ",scalar(@$set);
   say "smallest value: ",$set->[0];
   say "largest value: ",$set->[-1];

Given a list of integers, returns an array reference representing the
integer set.
The result is numerically sorted with duplicates removed.
The input array must only contain integers (signed integers, bigints,
objects that evaluate to integers, strings representing integers are all ok).
This "set form" is optimal for the set operations.

After the set is in this form, the size of the set is simply the length.
Similarly the set minimum and maximum are trivial.  All values in the output
will be typed as either native integers (IV or UV) or bigints.


=head2 setinsert

   my $s=[-10..-1,1..10];
   setinsert($s, 0);            # $s is now [-10..10]
   setinsert($s, [5,10,15,20]); # $s is now [-10..10,15,20]

Given two array references of integers in set form, inserts all elements
of the second set into the first set and returns the number of elements
that were inserted.

Given an array reference of integers in set form, followed by zero or more
integer scalars (possibly unordered and containing duplicates), inserts
all list values into the first set and returns the number of elements that
were inserted.  This is essentially the same as wrapping the list in
L</toset> but convenient and possibly more efficient.

This may be viewed as an in-place L</setunion>.

The one or two sets (array references) must be in set form (numerically
sorted with no duplicates) or the results are undefined.

=head2 setremove

   my $s=[-10..10];
   setremove($s, 0);            # $s is now [-10..-1,1..10]
   setremove($s, [5,10,15,20]); # $s is now [-10..-1,1..4,6..9]

Given two array references of integers in set form, removes all elements
of the second set from the first set and returns the number of elements
that were removed.

Given an array reference of integers in set form, followed by zero or more
integer scalars (possibly unordered and containing duplicates), removes
all list values from the first set and returns the number of elements that
were removed.  This is essentially the same as wrapping the list in
L</toset> but convenient and possibly more efficient.

This may be viewed as an in-place L</setminus>.

The one or two sets (array references) must be in set form (numerically
sorted with no duplicates) or the results are undefined.

=head2 setinvert

   my $s=[-10..10];
   setinvert($s, 0);            # $s is now [-10..-1,1..10]
   setinvert($s, [5,10,15,20]); # $s is now [-10..-1,1..4,6..9,15,20]

Given two array references of integers in set form, inverts the containment
status in the first set for each element of the second set.  That is, for
each element of the second set, inserts into the first set if not
an element, and removes from the first set if it is an element.

Given an array reference of integers in set form, followed by zero or more
integer scalars (possibly unordered and containing duplicates), does the
same as if the list was wrapped in L</toset>.

An integer value is returned indicating how many values were inserted,
minus the number of values deleted.

This may be viewed as an in-place L</setdelta>.

The one or two sets (array references) must be in set form (numerically
sorted with no duplicates) or the results are undefined.

=head2 setcontains

   my $has_element = setcontains( [-12,1..20], 15 );
   my $is_subset   = setcontains( [-12,1..20], [-12,5,10,15] );

Given two sets (array references of numerically sorted de-duplicated integers),
returns either 1 or 0 indicating whether the second argument
is a subset of the first set
(i.e. if all elements from the second argument are members of the first set).

Given a set and zero or more integers in any form (possibly unordered and can
contain duplicates), does the same as if the list was wrapped in L</toset>.

If the first array reference is not in set form (numerically sorted with no
duplicates, and no string forms), the result is undefined.  It is unlikely
to give a correct answer.  Use L</toset> to convert an arbitrary integer list
into set form.


=head2 setcontainsany

  # True if there is any intersection between the two sets
  my $intersects = setcontainsany($set1,$set2);
  my $has_one_of = setcontainsany( [-12,1..20], -14,0,1,100 );  # true

Given two sets (array references of numerically sorted de-duplicated integers),
returns either 1 or 0 indicating whether B<any> element of the second set
is an element of the first set.

Alternately, a set followed by a list of unordered integers will do the same,
as if the list was wrapped in L</toset>.

There is some functionality duplication, e.g. checking for disjoint sets
can be done with any of these:

  my $dj1 = set_is_disjoint($set1, $set2);
  my $dj2 = scalar(@{setintersect($set1, $set2)}) == 0;
  my $dj3 = !setcontainsany($set1, $set2);

This function B<requires> the array reference inputs be in set form or
the result is undefined.  In return it can be thousands of times faster
for large sets.


=head2 setbinop

  my $sumset = setbinop { $a + $b } [1,2,3], [2,3,4];  # [3,4,5,6,7]
  my $difset = setbinop { $a - $b } [1,2,3], [2,3,4];  # [-3,-2,-1,0,1]
  my $setsum = setbinop { $a + $b } [1,2,3];           # [2,3,4,5,6]

Given a code block and two array references containing integers,
treats them as integer sets and constructs a new set from applying
the cross product to the block.
If only one array reference is given, it will be used with itself.

The result will be in set form (numerically sorted, no duplicates).
The input sets are not aliased inside the block (modifying C<$a> and
C<$b> has no effect outside the block).

This corresponds to Pari's C<setbinop> function.
Our function uses B<much> less memory, as of Pari 2.18.1.

=head2 sumset

Given two array references of integers, treats them as integer sets and
returns the sumset as a set (a sorted de-duplicated array reference).

If only one array reference is given, it will be used for both.
It is common to see sumset applied to a single set.

This is equivalent to:

  my %r;  my @A=(2,4,6,8);  my @B=(3,5,7);
  forsetproduct { $r{vecsum(@_)}=undef; } \@A,\@B;
  my $sumset = [vecsort(keys %r)];

or

  my $sumset1 = setbinop { addint($a,$b) } [1,2,3];
  my $sumset2 = setbinop { addint($a,$b) } [1,2,3], [2,3,4];

In Mathematica one can use C<Total[Tuples[A,B],{2}]>.
In Pari/GP one can use C<setbinop((a,b)->a+b,X,Y)>.

=head2 setunion

Given exactly two array references of integers, treats them as sets and
returns the union as a set.
The returned set will have all elements that appear in either input set.

This is more efficient if the input is in set form
(numerically sorted, no duplicates).
The result will be in set form.

This corresponds to Pari's C<setunion> function,
Mathematica's C<Union> function, and
Sage's C<union> function on Set objects.

=head2 setintersect

  my $commonset = setintersect($set1,$set2);
  my $is_disjoint = 0 == @$commonset;  # scalar size of the intersection

Given exactly two array references of integers, treats them as sets and
returns the intersection as a set.
The returned set will have all elements that appear in both input sets.

This is more efficient if the input is in set form
(numerically sorted, no duplicates).
The result will be in set form.

This corresponds to Pari's C<setintersect> function,
Mathematica's C<Intersection> function, and
Sage's C<intersection> function on Set objects.

=head2 setminus

Given exactly two array references of integers, treats them as sets and
returns the difference as a set.
The returned set will have all elements that appear in the first set but
not in the second.

This is more efficient if the input is in set form
(numerically sorted, no duplicates).
The result will be in set form.

This corresponds to Pari's C<setminus> function,
Mathematica's C<Complement> function, and
Sage's C<difference> function on Set objects.

=head2 setdelta

Given exactly two array references of integers, treats them as sets and
returns the symmetric difference as a set.
The returned set will have all elements that appear in only one of the
two input sets.

This is more efficient if the input is in set form
(numerically sorted, no duplicates).
The result will be in set form.

This corresponds to Pari's C<setdelta> function,
Mathematica's C<SymmetricDifference> function, and
Sage's C<symmetric_difference> function on Set objects.

=head2 is_sidon_set

Given an array reference of integers, treats it as a set and returns 1
if it is a Sidon set (sometimes called Sidon sequence), and 0 otherwise.
To be a Sidon set, all elements must be non-negative and
all pair-wise sums a_i + a_j (i E<gt>= j) are unique.

All finite Sidon sets are Golomb rulers, and all Golomb rulers are Sidon.

=head2 is_sumfree_set

Given an array reference of integers, treats it as a set and returns 1
if it is a sum-free set, and 0 otherwise.
A sum-free set is one where no sum of two elements from the set is equal
to any element of the set.  That is, the set and its sumset are disjoint.

=head2 set_is_disjoint

Given two array references of integers, treats them as sets and
returns 1 if the sets have no elements in common, 0 otherwise.

This corresponds to Mathematica's C<DisjointQ> function.

=head2 set_is_equal

Given two array references of integers in set form,
returns 1 if the sets have all elements in common, 0 otherwise.

This function works even if the inputs are not sorted.  If they are sorted
(proper set form) then L</vecequal> can be used and is typically much faster.

=head2 set_is_subset

Given two array references of integers in set form,
returns 1 if the first set also contains all elements of the second set,
0 otherwise.

The L</setcontains> function can be used equivalently, and
does not require the second list to be in set form.

This corresponds to Mathematica's C<SubsetQ> function (is B a subset of A).

=head2 set_is_proper_subset

Given two array references of integers in set form,
returns 1 if the first set also contains all elements of the second set
but are not equal, 0 otherwise.
The size of the first set must be strictly larger than the second.

=head2 set_is_superset

Given two array references of integers in set form,
returns 1 if the second set also contains all elements of the first set,
0 otherwise.

The L</setcontains> function can be used equivalently
(with reversed arguments).

=head2 set_is_proper_superset

Given two array references of integers in set form,
returns 1 if the second set also contains all elements of the first set
but are not equal, 0 otherwise.
The size of the second set must be strictly larger than the first.

=head2 set_is_proper_intersection

Given two array references of integers in set form,
returns 1 if the two sets have at least one element in common,
and each of the two sets have at least one element not present
in the other set.  Returns 0 otherwise.


=head2 todigits

  say "product of digits of n: ", vecprod(todigits($n));

Given an integer C<n>, return an array of digits of C<|n|>.  An optional
second integer argument specifies a base (default 10).  For example,
given a base of 2, this returns an array of binary digits of C<n>.
An optional third argument specifies a length for the returned array.
The result will be either have upper digits truncated or have leading
zeros added.  This is most often used with base 2, 8, or 16.

The values returned may be read-only.  C<todigits(0)> returns an empty array.
The base must be at least 2, and is limited to an int.  Length must be
at least zero and is limited to an int.

This corresponds to Pari's C<digits> and C<binary> functions, and
Mathematica's C<IntegerDigits> function.

=head2 todigitstring

  # arguments are:  input integer, base (optional), truncate (optional)
  say "decimal 456 in hex is ", todigitstring(456, 16);
  say "last 4 bits of $n are: ", todigitstring($n, 2, 4);

Similar to L</todigits> but returns a string.
For bases E<lt>= 10, this is equivalent to joining the array returned
by L</todigits>.

The first argument C<n> is the input integer.  The sign is ignored.
If no other arguments are given, this just returns the string of C<n>.
An optional second argument is the base C<base> which must be between 2 and 36.
No prefix such as "0x" will be added, and all bases over 9 use lower case
C<a> to C<z>.

An optional third argument C<k> requires the result to be exactly C<k> digits.
This truncates to the last C<k> digits if the result has C<k> or fewer digits,
or zero extends if the result has more digits.

This corresponds to Mathematica's C<IntegerString> function.

=head2 fromdigits

  say "hex 1c8 in decimal is ", fromdigits("1c8", 16);
  say "Base 3 array to number is: ", fromdigits([0,1,2,2,2,1,0],3);

This takes either a string or array reference, and an optional base
(default 10).  With a string, each character will be interpreted as a
digit in the given base, with both upper and lower case denoting
values 11 through 36.  With an array reference, the values indicate
the entries in that location, and values larger than the base are
allowed (results are carried).  The result is a number (either a
native integer or a bigint).

This corresponds to Pari's C<fromdigits> function and
Mathematica's C<FromDigits> function.

=head2 tozeckendorf

  say tozeckendorf(24);                     #  "1000100"
  say fromdigits(tozeckendorf(24),2);       #  68

Given a non-negative integer C<n>, return the Zeckendorf representation as
a binary string.  This represents C<n> as a sum of nonconsecutive
Fibonacci numbers.
Each set bit indicates summing the corresponding Fibonacci number,
e.g. 24 = 21+3 = F(8)+F(4).
F(0)=0 and F(1)=1 are not used.
This is sometimes also called Fibbinary or the Fibonacci base.

The restriction that consecutive values are not used ("11" cannot appear)
is required to create a unique mapping to the positive integers.
A simple greedy algorithm suffices to construct the encoding.

    say reverse(tozeckendorf($_)).'1'  for  1..20

shows the first twenty Fibonacci C1 codes (Fraenkel and Klein, 1996).
This is an example of a self-synchronizing variable length code.

This corresponds to Mathematica's C<ZeckendorfRepresentation[n]> function.
Also see L<Math::NumSeq::Fibbinary> and L<Data::BitStream::Code::Fibonacci>.

=head2 fromzeckendorf

  say fromzeckendorf("1000100");            #  24
  say fromzeckendorf(todigitstring(68,2));  #  24

Given a binary string in Zeckendorf representation, return the corresponding
integer.  The string may not contain anything other than the characters
C<0> and C<1>, and must not contain C<11>.  The resulting number is the sum
of the Fibonacci numbers in the position starting from the right
(The Fibonacci index is offset by two, as F(0)=0 and F(1)=1 are not used).

=head2 sumdigits

  # Sum digits of primes to 1 million.
  my $s=0; forprimes { $s += sumdigits($_); } 1e6; say $s;

Given an input C<n>, return the sum of the digits of C<n>.  Any non-digit
characters of C<n> are ignored (including negative signs and decimal points).
This is similar to the command C<vecsum(split(//,$n))> but faster,
allows non-positive-integer inputs, and can sum in other bases.

An optional second argument indicates the base of the input number.
This defaults to 10, and must be between 2 and 36.  Any character that is
outside the range C<0> to C<base-1> will be ignored.

If no base is given and the input number C<n> begins with C<0x> or C<0b>
then it will be interpreted as a string in base 16 or 2 respectively.

Regardless of the base, the output sum is a decimal number.

This is similar but not identical to Pari's C<sumdigits> function from
version 2.8 and later.  The Pari/GP function always takes the input as
a decimal number, uses the optional base as a base to first convert to,
then sums the digits.  This can be done with either
C<vecsum(todigits($n, $base))> or C<sumdigits(todigitstring($n,$base))>.
C<Math::BigInt> version 1.999818 has a similar C<digitsum> function.

=head2 valuation

  say "$n is divisible by 2 ", valuation($n,2), " times.";

Given integer C<n> and non-negative integer C<k>, returns the number
of times C<n> is divisible by C<k>.
This is a very limited version of the algebraic valuation -- here
it is just applied to integers.

C<k> must be greater than 1.
C<|n|> is used, C<|n| = 0> returns undef, and C<|n| = 1> returns zero.

This corresponds to Pari and SAGE's C<valuation> function.

=head2 hammingweight

Given an integer C<n>, returns the binary Hamming weight of C<abs(n)>.  This
is also called the population count, and is the number of 1s in the binary
representation.  This corresponds to Pari's C<hammingweight> function for
C<t_INT> arguments.

=head2 is_square_free

  say "$n has no repeating factors" if is_square_free($n);

Given integer C<n>, returns 1 if C<|n|> has no repeated factor.

=head2 is_cyclic

Given integer C<n>, returns 1 if C<n> is positive and cyclic in the number
theory sense, and returns 0 otherwise.
A cyclic number C<n> has only one group of order C<n>.
C<n> and C<φ(n)> are relatively prime.

This is the L<OEIS series A003277|http://oeis.org/A003277>.

=head2 is_carmichael

  for (1..1e6) { say if is_carmichael($_) } # Carmichaels under 1,000,000

Given an integer C<n>, returns 1 if C<n> is positive and
a Carmichael number, and returns 0 otherwise.
These are composites that satisfy C<b^(n-1) ≡ 1 mod n> for all
C<< 1 < b < n >> relatively prime to C<n>.
Alternately Korselt's theorem says these are composites such that C<n> is
square-free and C<p-1> divides C<n-1> for all prime divisors C<p> of C<n>.

For inputs larger than 50 digits after removing very small factors, this
uses a probabilistic test since factoring the number could take unreasonably
long.  The first 150 primes are used for testing.  Any that divide C<n> are
checked for square-free-ness and the Korselt condition, while those that do
not divide C<n> are used as the pseudoprime base.  The chances of a
non-Carmichael passing this test are less than C<2^-150>.

This is the L<OEIS series A002997|http://oeis.org/A002997>.

=head2 is_quasi_carmichael

Given an integer C<n>, returns 0 if C<n> is negative or not
a quasi-Carmichael number, and returns the number of bases otherwise.
These are square-free composites that satisfy
C<p+b> divides C<n+b> for all prime factors C<p> of C<n> and for one or
more non-zero integer C<b>.

This is the L<OEIS series A257750|http://oeis.org/A257750>.

=head2 is_semiprime

Given an integer C<n>, returns 1 if C<n> is positive and
a semiprime, and returns 0 otherwise.
A semiprime is the product of exactly two primes.

The boolean result is the same as C<scalar(factor(n)) == 2>, but this
function performs shortcuts that can greatly speed up the operation.

=head2 is_almost_prime

  say is_almost_prime(6,2169229601);  # True if n has exactly 6 factors

Given non-negative integers C<k> and C<n>, returns 1 if C<n> has
exactly C<k> prime factors, and 0 otherwise.
With C<k=1>, this is a standard primality test.
With C<k=2>, this is the same as L</is_semiprime>.

Functionally identical but possibly faster than C<prime_bigomega(n) == k>.

=head2 is_omega_prime

  say is_omega_prime(6,2169229601);  # True if n has 6 distinct factors

Given non-negative integers C<k> and C<n>, returns 1 if C<n> has
exactly C<k> distinct prime factors (not counting multiplicity), and 0 otherwise.
With C<k=1>, this is the same as L</is_prime_power>.

Functionally identical but possibly faster than C<prime_omega(n) == k>.

=head2 is_chen_prime

Given non-negative integer C<n> return 1 if C<n> is a Chen prime.  That is,
if C<n> is prime and C<n+2> is either a prime or semi-prime.

=head2 is_fundamental

Given an integer C<d>, returns 1 if C<d> is a fundamental discriminant,
0 otherwise.  We consider 1 to be a fundamental discriminant.

This is the L<OEIS series A003658|http://oeis.org/A003658> (positive) and
L<OEIS series A003657|http://oeis.org/A003657> (negative).

This corresponds to Pari's C<isfundamental> function.

=head2 is_totient

Given an integer C<n>, returns 1 if there exists an integer C<x> where
C<euler_phi(x) == n>.

This corresponds to Pari's C<istotient> function, though without the
optional second argument to return an C<x>.  L<Math::NumSeq::Totient>
also has a similar function.

Also see L</inverse_totient> which gives the count or list of values that
produce a given totient.  This function is more efficient than getting the
full count or list.

=head2 is_pillai

Given a non-negative integer C<n>, if there exists a C<v> where C<v! % n == n-1>
and C<n % v != 1>, then the least C<v> is returned.  Otherwise 0.

For n prime, non-zero return values give
L<OEIS series A063980|http://oeis.org/A063980>.
The non-zero values returned produce
L<OEIS series A063828|http://oeis.org/A063828>.


=head2 is_polygonal

Given an integer C<x> and a positive integer C<s> greater than 2,
return 1 if x is an s-gonal number, and return 0 otherwise.

If a third argument is present, it must be a scalar reference.  It will be
set to n if x is the nth s-gonal number.  If the function returns 0, then
it will be unchanged.

This corresponds to Pari's C<ispolygonal> function.

=head2 is_congruent_number

Given a non-negative integer C<n>, returns 1 if C<n> is the area of a
rational right triangle, and 0 otherwise.

This function answers the B<congruent number problem> using Tunnell's theorem
which relies on the Birch Swinnerton-Dyer conjecture.  It uses an extensive
filter for known non-congruent families, including the works of
Bastien (1915), Lagrange (1974), Monsky (1990), Serf (1991),
Iskra (1996), Feng (1996), Reinholz et al. (2013),
Cheng and Guo (2018 and 2019), Das and Saikia (2020), and Evink (2021).

=head2 cornacchia

Given non-negative integers C<d> and C<n>, finds solutions C<(x,y)> to the
equation C<x^2 + d y^2 = n>.  C<undef> is returned if no solution exists.

In the case of C<n> a prime, this is done using Cornacchia's algorithm.

For non-prime C<n>, we use a combination of Cornacchia-Smith on all roots,
as well as a loop to find solutions in the harder cases.  This means we
will always return a solution.

There will often be multiple solutions, but only one is returned.


=head2 contfrac

  my @CF = contfrac(415,93);
  # CF = (4,2,6,7)  =>  4+(1/(2+1/(6+1/7))) = 415/93
  #                     ^     ^    ^   ^

Given an integer C<n> and a positive integer C<d>,
returns a list with the simple continued fraction representation
of the rational C<n / d>.

This corresponds to a subset of Pari's C<contfrac> function,
Mathematica's C<ContinuedFraction[n/d]> function,
and Sage's C<continued_fraction> function.

=head2 from_contfrac

  my($N,$D) = from_contfrac(4,2,6,7);  # N = 415, D = 93

Given an array of integers representing the simple continued fraction,
returns the rational C<n / d> as two integers.

The first input value represents the whole part, and may be zero or negative.
All successive input values must be non-negative and non-zero.

This corresponds to a subset of Pari's C<contfracpnqn> function,
Mathematica's C<FromContinuedFraction[list]> function,
and one value of Sage's C<convergent(n)> method.

=head2 next_calkin_wilf

  ($n,$d) = next_calkin_wilf($n,$d);

Given two positive coprime integers C<n> and C<d> representing
the rational C<n / d>, returns the next value in the breadth-first
traversal of the Calkin-Wilf tree of rationals as a two-element list.

The Calkin-Wilf tree has an entry for all positive rationals in lowest
form, with each one appearing only once.  While it is not a binary search
tree over the positive rationals like the Stern-Brocot tree, it is
more efficient to traverse in both depth and breadth order.

This corresponds to Julia's Nemo C<next_calkin_wilf> function.

This can efficiently iterate through
L<OEIS series A002487|http://oeis.org/A002487>.

=head2 next_stern_brocot

  ($n,$d) = next_stern_brocot($n,$d);

Given two positive coprime integers C<n> and C<d> representing
the rational C<n / d>, returns the next value in the breadth-first
traversal of the Stern-Brocot tree of rationals as a two-element list.

The Stern-Brocot tree has an entry for all positive rationals in lowest
form, with each one appearing only once.
Read left-to-right on each row, the numbers appear in ascending order.
It is a binary search tree over the positive rationals (this was exactly
Brocot's motivation).
It is not as efficient as L</next_calkin_wilf>.

This produces L<OEIS series A007305|http://oeis.org/A007305> (numerators)
and L<OEIS series A047679|http://oeis.org/A047679> (denominators).

=head2 calkin_wilf_n

  my $idx = calkin_wilf_n($n,$d);

Given two positive coprime integers C<n> and C<d> representing
the rational C<n / d>, returns the index in the breadth-first
traversal of the Calkin-Wilf tree of rationals.

This corresponds to the C<xy_to_n> method
in L<Math::PlanePath::RationalsTree> with C<tree_type => 'CW'>.

=head2 stern_brocot_n

  my $idx = stern_brocot_n($n,$d);

Given two positive coprime integers C<n> and C<d> representing
the rational C<n / d>, returns the index in the breadth-first
traversal of the Stern-Brocot tree of rationals.

This corresponds to the C<xy_to_n> method
in L<Math::PlanePath::RationalsTree> with C<tree_type => 'SB'>.

=head2 nth_calkin_wilf

  ($n,$d) = nth_calkin_wilf($idx);

Given a positive integer C<i>, returns the rational in the
corresponding index in
the breadth-first traversal of the Calkin-Wilf tree of rationals.

This corresponds to the C<n_to_xy> method
in L<Math::PlanePath::RationalsTree> with C<tree_type => 'CW'>.

=head2 nth_stern_brocot

  ($n,$d) = nth_stern_brocot($idx);

Given a positive integer C<i>, returns the rational in the
corresponding index in
the breadth-first traversal of the Stern-Brocot tree of rationals.

This corresponds to the C<n_to_xy> method
in L<Math::PlanePath::RationalsTree> with C<tree_type => 'SB'>.

=head2 nth_stern_diatomic

  $n = nth_stern_diatomic($idx);

Given a non-negative integer C<i>, returns the C<i>-th Stern diatomic number.
This is sometimes called C<fusc(i)> (Dijkstra), Stern's diatomic series,
or the Stern-Brocot sequence.  The latter can be easily confused with the
Stern-Brocot tree.

This corresponds to Sidef's C<fusc> function.  See also L</next_calkin_wilf>
where the sequence of numerators generates this sequence.

This produces L<OEIS series A002487|http://oeis.org/A002487>.

=head2 farey

  #    F[3] = 0/1  1/3  1/2  2/3  1/1
  #
  say scalar farey(3);   #  5
  my @F3 = farey(3);     #  ([0,1], [1,3], [1,2], [2,3], [1,1])
  my $F33 = farey(3,3);  #  [2/3] = $F3[3]
  # Print the list in readable form
  say join " ",map { join "/",@$_ } farey(3);

Given a single positive integer C<n> returns the Farey sequence of order C<n>.
In scalar context, returns the length without computing terms.
In array context, returns a list with each rational as a 2-entry array
reference.

Given two values: a positive integer C<n> and a non-negative integer C<k>,
returns the C<k-th> entry of the order C<n> Farey sequence.  The index
starts at zero so it matches using the full list as an array.
If C<k> is larger than the number of entries, undef is returned.

This corresponds to Mathematica's C<FareySequence> function (their
two argument version is 1-based rather than 0-based).

The lengths are L<OEIS series A005728|http://oeis.org/A005728>.
The numerators are L<OEIS series A006842|http://oeis.org/A006842>.
The denominators are L<OEIS series A006843|http://oeis.org/A006843>.

=head2 next_farey

  my $next = next_farey(9,[5,9]);  # returns [4,7]

Given a positive integer C<n> and a 2-element array reference containing
a non-negative integer C<p> and a positive integer C<q>, returns the next
rational appearing after C<p/q> in the order C<n> Farey sequence.
Returns undef if C<p/q> is greater than or equal to one.

=head2 farey_rank

  my $rank = farey_rank(9,[5,9]);  # returns 15

Given a positive integer C<n> and a 2-element array reference containing
a non-negative integer C<p> and a positive integer C<q>, returns the number
of rationals less than C<p/q> in the order C<n> Farey sequence.
The given fraction does not need to be an entry in the sequence, nor does
it need to be in reduced form.

A unit fraction will return the totient sum of C<n>.  Any fraction greater
than one will return the length of the order C<n> sequence, as expected.

Many OEIS sequences can be produced from this, including
L<OEIS series A005728|http://oeis.org/A005728> (E<lt>= 1),
L<OEIS series A049806|http://oeis.org/A049806> (E<lt>= 1/2),
L<OEIS series A049807|http://oeis.org/A049807> (E<lt>= 1/3),
L<OEIS series A049808|http://oeis.org/A049808> (E<lt>= 1/4),
...,
L<OEIS series A049805|http://oeis.org/A049805> (E<lt>= 1/k),


=head2 prime_bigomega

  say "$n has ", prime_bigomega($n), " total factors";

Given a non-negative integer C<n>, returns Ω(|n|), the prime Omega function.
This is the total number of prime factors of C<n> including multiplicities.
The result is identical to C<scalar(factor($n))>.
The return value is a read-only constant.

This corresponds to Pari's C<bigomega> function
and Mathematica's C<PrimeOmega[n]> function.

=head2 prime_omega

  say "$n has ", prime_omega($n), " distinct factors";

Given a non-negative integer C<n>, returns ω(|n|), the prime omega function.
This is the number of distinct prime factors of C<n>.
The result is identical to C<scalar(factor_exp($n))>.
The return value is a read-only constant.

This corresponds to Pari's C<omega> function
and Mathematica's C<PrimeNu[n]> function.

=head2 moebius

  say "$n is square free" if moebius($n) != 0;
  $sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
  say "Mertens(2000) = ", vecsum(moebius(0,2000));

Given a single integer C<n>, returns μ(|n|), the Möbius function
(also known as the Moebius, Mobius, or MoebiusMu function).
This function is 1 if C<n = 1>, 0 if C<n> is not square-free
(i.e. C<n> has a repeated factor), and C<(-1)^t> if C<n> is a product
of C<t> distinct primes.
This is an important function in prime number theory.  Like SAGE, we define
C<moebius(0) = 0> for convenience.

If given two integers C<low> and C<high>, they define a range, and the
function returns an array with the value of the Möbius function
for every C<|n|> from C<low> to C<high> inclusive.
Large values of C<high> will result in a lot of
memory use.  The algorithm used for ranges is Deléglise and Rivat (1996)
algorithm 4.1, which is a segmented version of Lioen and van de Lune (1994)
algorithm 3.2.

Negative ranges are possible, e.g. C<moebius(-30,-20)> will return
C<moebius(|n|)> for -30, -29, -28, ..., -20.

The return values are read-only constants.  This should almost never come up,
but it means trying to modify aliased return values will cause an
exception (modifying the returned scalar or array is fine).


=head2 mertens

  say "Mertens(10M) = ", mertens(10_000_000);   # = 1037

Given a non-negative integer C<n>, return M(n), the Mertens function.
This function is defined as C<sum(moebius(1..n))>, but calculated more
efficiently for large inputs.  For example, computing Mertens(100M) takes:

   time    approx mem
     0.01s     0.1MB   mertens(100_000_000)
     1.3s    880MB     vecsum(moebius(1,100_000_000))
    16s        0MB     $sum += moebius($_) for 1..100_000_000

The summation of individual terms via factoring is quite expensive in time,
though uses O(1) space.  Using the range version of moebius is much faster,
but returns a 100M element array which, even though they are shared constants,
is not good for memory at this size.
In comparison, this function will generate the equivalent output
via a sieving method that is relatively memory frugal and very fast.
The current method is a simple C<n^1/2> version of Deléglise and Rivat (1996),
which involves calculating all moebius values to C<n^1/2>, which in turn will
require prime sieving to C<n^1/4>.

Various algorithms exist for this, using differing quantities of μ(n).  The
simplest way is to efficiently sum all C<n> values.  Benito and Varona (2008)
show a clever and simple method that only requires C<n/3> values.  Deléglise
and Rivat (1996) describe a segmented method using only C<n^1/3> values.  The
current implementation does a simple non-segmented C<n^1/2> version of their
method.  Kuznetsov (2011) gives an alternate method that he indicates is even
faster.  Helfgott and Thompson (2020) give a fast method based on advanced
prime count algorithms.


=head2 euler_phi

  say "The Euler totient of $n is ", euler_phi($n);

Given a single integer C<n>, returns φ(n), the Euler totient function
(also called Euler's phi or phi function).
This is an arithmetic function which counts the number of positive
integers less than or equal to C<n> that are relatively prime to C<n>.

Given the definition used, C<euler_phi> will return 0 for all
C<< n < 1 >>.  This follows the logic used by SAGE.  Mathematica and Pari
return C<euler_phi(-n)> for C<< n < 0 >>.  Mathematica returns 0 for C<n = 0>,
Pari pre-2.6.2 raises an exception, and Pari 2.6.2 and newer returns 2.

If called with two integer arguments C<low> and C<high>, they define
an inclusive range.
The function returns a list with the totient of every n from low to high
inclusive.

=head2 inverse_totient

In array context, given a non-negative integer C<n>, returns the complete list
of values C<x> where C<euler_phi(x) = n>.  This can be a memory intensive
operation if there are many values.

In scalar context, returns just the count of values.  This is faster
and uses substantially less memory.  The list/scalar distinction is
similar to L</factor> and L</divisors>.

This roughly corresponds to the Maple function C<InverseTotient>, and the
hidden Mathematica function C<EulerPhiInverse>.  The algorithm used is
from Max Alekseyev (2016).

=head2 jordan_totient

  say "Jordan's totient J_$k($n) is ", jordan_totient($k, $n);

Given non-negative integers C<k> and C<n>, returns Jordan's k-th totient
function for C<n>.
Jordan's totient is a generalization of Euler's totient, where
  C<jordan_totient(1,$n) == euler_phi($n)>
This counts the number of k-tuples less than or equal to n that form a coprime
tuple with n.  As with C<euler_phi>, 0 is returned for all C<< n < 1 >>.
This function can be used to generate some other useful functions, such as
the Dedekind psi function, where C<psi(n) = J(2,n) / J(1,n)>.

=head2 sumtotient

Given a non-negative integer C<n>,
returns the summatory Euler totient function.
This function is defined as C<sum(euler_phi(1..n))>, but calculated
much more efficiently.

A sub-linear time recursion is implemented, using O(n^2/3) memory.
Memory use is restricted so growth becomes approximately linear above C<10^13>.

This is L<OEIS series A002088|http://oeis.org/A002088>.


=head2 ramanujan_sum

Given two non-negative integers C<k> and C<n>, returns Ramanujan's sum.
This is the sum of the nth powers of the primitive k-th roots of unity.

Note this is not related to Ramanujan summation for divergent series.


=head2 exp_mangoldt

  say "exp(lambda($_)) = ", exp_mangoldt($_) for 1 .. 100;

Given a non-negative integer C<n>, returns EXP(Λ(n)), the exponential
of the Mangoldt function (also known as von Mangoldt's function).
The Mangoldt function is equal to log p if n is prime or a power of a prime,
and 0 otherwise.  We return the exponential so all results are integers.
Hence the return value for C<exp_mangoldt> is:

   p   if n = p^m for some prime p and integer m >= 1
   1   otherwise.


=head2 liouville

Given a non-negative integer C<n>, returns λ(n), the Liouville function.
This is -1 raised to Ω(n) (the total number of prime factors).

This corresponds to Mathematica's C<LiouvilleLambda[n]> function.
It can be computed in Pari/GP as C<(-1)^bigomega(n)>.

=head2 sumliouville

Given a non-negative integer C<n>, returns L(n),
the summatory Liouville function.
This function is defined as C<sum(liouville(1..n))>, but calculated
much more efficiently.

There are a number of relations to the L</mertens> function.

This is L<OEIS series A002819|http://oeis.org/A002819>.


=head2 chebyshev_theta

  say chebyshev_theta(10000);

Given a non-negative integer C<n>, returns θ(n),
the first Chebyshev function.
This is the sum of the logarithm of each prime where C<< p <= n >>.
Effectively:

  my $s = 0;  forprimes { $s += log($_) } $n;  return $s;

but computed more efficiently and accurately.

=head2 chebyshev_psi

  say chebyshev_psi(10000);

Given a non-negative integer C<n>, returns ψ(n),
the second Chebyshev function.
This is the sum of the logarithm of each prime power where C<< p^k <= n >>
for an integer k.
Effectively:

  my $s = 0;  for (1..$n) { $s += log(exp_mangoldt($_)) }  return $s;

but computed more efficiently and accurately.
We compute it as a Neumaier sum from C<k = 1 .. floor(log2(n))> of
C<chebyshev_theta(n^(1/k))>.

=head2 divisor_sum

  say "Sum of divisors of $n:", divisor_sum( $n );
  say "sigma_2($n) = ", divisor_sum($n, 2);
  say "Number of divisors: sigma_0($n) = ", divisor_sum($n, 0);

Given a single non-negative integer C<n>, returns the sum of the
divisors of C<n>, including 1 and itself.  We return 0 for C<n=0>.

An optional second non-negative integer C<k> may be given, indicating
the sum should use the C<k-th> powers of the divisors.

This is known as the sigma function (see Hardy and Wright section 16.7).
The API is identical to Pari/GP's C<sigma> function, and not dissimilar to
Mathematica's C<DivisorSigma[k,n]> function.
This function is useful for calculating things like aliquot sums, abundant
numbers, perfect numbers, etc.

With various C<k> values, the results are the OEIS sequences
L<OEIS series A000005|http://oeis.org/A000005> (C<k=0>, number of divisors),
L<OEIS series A000203|http://oeis.org/A000203> (C<k=1>, sum of divisors),
L<OEIS series A001157|http://oeis.org/A001157> (C<k=2>, sum of squares of divisors),
L<OEIS series A001158|http://oeis.org/A001158> (C<k=3>, sum of cubes of divisors),
etc.

The second argument may also be a code reference, which is called for each
divisor and the results are summed.  This allows computation of other
functions, but will be less efficient than using the numeric second argument.
This corresponds to Pari/GP's C<sumdiv> function.

An example of the 5th Jordan totient (OEIS A059378):

  divisor_sum( $n, sub { my $d=shift; $d**5 * moebius($n/$d); } );

though we have a function L</jordan_totient> which is more efficient.

For numeric second arguments (sigma computations), the result will be a bigint
if necessary.  For the code reference case, the user must take care to return
bigints if overflow will be a concern.


=head2 ramanujan_tau

Given an integer C<n>, returns the value of Ramanujan's tau function.
The result is a signed integer.  Zero is returned for negative C<n>.
This corresponds to Pari v2.8's C<tauramanujan> function and
Mathematica's C<RamanujanTau> function.

This currently uses a simple method based on divisor sums, which does
not have a good computational growth rate.  Pari's implementation uses
Hurwitz class numbers and is more efficient for large inputs.


=head2 primorial

  $prim = primorial(11); #        11# = 2*3*5*7*11 = 2310

Given a non-negative integer C<n>, returns the primorial C<n#>,
defined as the
product of the prime numbers less than or equal to C<n>.  This is the
L<OEIS series A034386|http://oeis.org/A034386>: primorial numbers second
definition.

  primorial(0)  == 1
  primorial($n) == pn_primorial( prime_count($n) )

The result will be a L<Math::BigInt> object if it is larger than the native
bit size.

Be careful about which version (C<primorial> or C<pn_primorial>) matches the
definition you want to use.  Not all sources agree on the terminology, though
they often give a clear definition of which of the two versions they mean.
OEIS, Wikipedia, and Mathworld are all consistent, and these functions should
match that terminology.  This function should return the same result as the
C<mpz_primorial_ui> function added in GMP 5.1.


=head2 pn_primorial

  $prim = pn_primorial(5); #      p_5# = 2*3*5*7*11 = 2310

Given a non-negative integer C<n>, returns the primorial number C<p_n#>,
defined as
the product of the first C<n> prime numbers (compare to the factorial, which
is the product of the first C<n> natural numbers).  This is the
L<OEIS series A002110|http://oeis.org/A002110>: primorial numbers first
definition.

  pn_primorial(0)  == 1
  pn_primorial($n) == primorial( nth_prime($n) )

The result will be a L<Math::BigInt> object if it is larger than the native
bit size.


=head2 consecutive_integer_lcm

  $lcm = consecutive_integer_lcm($n);

Given a non-negative integer C<n>, returns the least common multiple of all
integers from 1 to C<n>.  This can be done by manipulation of the primes up
to C<n>, resulting in much faster and memory-friendly results than using
a factorial.

This is L<OEIS series A003418|http://oeis.org/A003418>.
Matching that series, we define C<consecutive_integer_lcm(0) = 1>.

=head2 partitions

Given an integer C<n>, returns the partition function C<p(n)>.
If C<n> is negative, 0 is returned.
This is the number of ways of writing the integer C<n> as a sum of positive
integers, without restrictions.

This corresponds to Pari's C<numbpart>
function and Mathematica's C<PartitionsP> function.  The values produced
in order are L<OEIS series A000041|http://oeis.org/A000041>.

This uses a combinatorial calculation, which means it will not be very
fast compared to Pari, Mathematica, or FLINT which use the Rademacher
formula using multi-precision floating point.  In 10 seconds:

            70    Integer::Partition
            90    MPU forpart { $n++ }
        15_000    MPU pure Perl partitions
       280_000    MPU GMP partitions
    35_000_000    Pari 2.6 numbpart
   500_000_000    Jonathan Bober's partitions_c.cc v0.6
 1_400_000_000    Pari 2.8 numbpart

If you want the enumerated partitions, see L</forpart>.


=head2 lucky_numbers

Given a single 32-/64-bit non-negative integer C<n>,
returns an array reference of values up to the input C<n> (inclusive)
which remain after the lucky number sieve originally defined by
Gardiner, Lazarus, Metropolis, and Ulam.
This is L<OEIS series A000959|http://oeis.org/A000959>.

If given two non-negative integers C<lo> and C<hi>, returns sieve results
between the two ranges inclusive.  This is identical to the above but does
not include any numbers less than C<lo>.  Currently there is very little
time savings, but it does use less memory.

A surprising number of asymptotic properties of the primes are shared
with this sieve, though the resulting sets are quite different.

There is no current algorithm for efficiently sieving a segment, though
the method used here is orders of magnitude faster than those linked
on OEIS as of early 2023.
CPU time growth is similar to prime sieving, about C<n log n>.
Memory use is linear with size and uses about C<n/25> bytes for the
internal sieve.

=head2 is_lucky

Given an integer C<n>, returns C<1> if C<n> is included in the
set of lucky numbers and returns C<0> otherwise.
The process used is analogous to trial division using the lucky
numbers less than C<n/log(n)>.
For inputs not quickly discarded, the performance is essentially
the same as generating the nth lucky number nearest to the input.

=head2 lucky_count

Given a single non-negative integer C<n>, returns the count of lucky
numbers less than or equal to C<n>.
If given two non-negative integers C<lo> and C<hi>, returns the count
of lucky numbers between C<lo> and C<hi> inclusive.

=head2 lucky_count_approx

Given a single non-negative integer C<n>, quickly returns a
good estimate of the count of lucky numbers less than or equal to C<n>.

=head2 lucky_count_lower

Given a single non-negative integer C<n>, quickly returns a
lower bound of the count of lucky numbers less than or equal to C<n>.
The actual count will always be greater than or equal to the result.

=head2 lucky_count_upper

Given a single non-negative integer C<n>, quickly returns an
upper bound of the count of lucky numbers less than or equal to C<n>.
The actual count will always be less than or equal to the result.

=head2 nth_lucky

Given a non-negative integer C<n>, returns the C<n>-th lucky number.
This is done by sieving lucky numbers to C<n> then performing
a reverse calculation to determine the value at the nth position.
This is much more efficient than generating all the lucky numbers
up to the nth position, but is much slower than L</nth_prime>.

=head2 nth_lucky_approx

Given a single non-negative integer C<n>, quickly returns a
good estimate of the C<n>-th lucky number.

=head2 nth_lucky_lower

Given a single non-negative integer C<n>, quickly returns a
lower bound of the C<n>-th lucky number.
The actual value will always be greater than or equal to the result.

=head2 nth_lucky_upper

Given a single non-negative integer C<n>, quickly returns an
upper bound of the C<n>-th lucky number.
The actual value will always be less than or equal to the result.


=head2 minimal_goldbach_pair

Given a single non-negative integer C<n>, returns the smallest prime C<p>
such that C<p + q = n> and both C<p> and C<q> are primes.
Only the single value C<p> is returned, with C<q = n-p> and C<< p <= q >>.
Both C<p> and C<q> are prime.

C<undef> is returned if no such C<p> exists.  This will happen for values
less than C<4> and for all odd C<n> where C<n != 2+q> for a prime C<q>.
The Goldbach Conjecture famously states that a C<p> exists for
all even C<n> greater than C<2>.

This function is reasonably fast even for larger values of C<n> as it can
terminate after the first pair is found.  On Macbook M1, average time is
under 1 microsecond for 32-bit even inputs, under 10 microseconds for 64-bit
even inputs, and 1 millisecond for 105 bit even inputs.

=head2 goldbach_pair_count

Given a single non-negative integer C<n>, returns the number of pairs of
primes C<p> and C<q> where C<< p <= q >> and C<< p + q = n >>.

If no such pairs exist, C<0> is returned.

=head2 goldbach_pairs

Given a single non-negative integer C<n>, returns a list containing each C<p>
for all prime pairs C<p> and C<q> where C<< p <= q >> and C<p + q = n>.
The number of elements returned is the same as L</goldbach_pair_count>.

If no such pairs exist, an empty list is returned.


=head2 is_happy

Given a single non-negative integer C<n>, returns the number of iterations
required for the map of sum of squared base-10 digits to converge to C<1>,
or C<0> if it does not converge to the value C<1>.

This returns the height using the OEIS A090425 definition of height, which is
zero for non-happy numbers, 1 for C<n=1>, 2 for numbers that produce 1 after
a single iteration, etc.
This is one more than the definitions used in many papers
(e.g. Cai and Zhou 2008) where C<n=1> is considered to have height 0.

An optional base and exponent may be given (default base 10 exponent 2).
The base must be between 2 and 36, and the exponent between 0 and 10.
The input C<n> is read as a decimal number, so giving input such as "1001"
will be treated as the decimal C<1001> regardless of base.

With base 10 and exponent 2,
non-zero values produce L<OEIS series A007770|http://oeis.org/A007770>.
The values themselves produce L<OEIS series A090425|http://oeis.org/A090425>.



=head2 is_smooth

  my $is_23_smooth = is_smooth($n, 23);

Given two non-negative integer inputs C<n> and C<k>,
returns C<1> if C<|n|> is C<k>-smooth, and C<0> otherwise.
This uses the OEIS definition: Returns true if no prime factors
of C<n> are larger than C<k>.

The values for C<n=0> and C<n=1> use the definition along with noting
that C<factor(0)> returns 0 and C<factor(1)> returns an empty list.

The result is identical to:

  sub is_smooth { my($n,$k)=@_; return 0+(vecnone { $_ > $k } factor($n)); }

but shortcuts are taken to avoid fully factoring if possible.

This corresponds to Mathematica's C<SmoothIntegerQ[n]> resource function.

=head2 is_rough

  my $is_23_rough = is_rough($n, 23);

Given two non-negative integer inputs C<n> and C<k>,
returns C<1> if C<|n|> is C<k>-rough, and C<0> otherwise.
This uses the OEIS definition: Returns true if no prime factors
of C<n> are smaller than C<k>.

The values for C<n=0> and C<n=1> use the definition along with noting
that C<factor(0)> returns 0 and C<factor(1)> returns an empty list.

The result is identical to:

  sub is_rough { my($n,$k)=@_; return 0+(vecnone { $_ < $k } factor($n)); }

but shortcuts are taken to avoid fully factoring if possible.


=head2 is_powerful

  my $all_factors_cubes_or_higher = is_powerful($n, 3);

Given an integer C<n> and an optional non-negative integer C<k>,
returns C<1> if C<n> is C<k>-powerful, and C<0> otherwise.
If C<k> is omitted, C<k=2> is used.

A k-powerful number is a positive integer where all prime factors appear
at least C<k> times.
All positive integers are therefore 0- and 1-powerful.
C<n=1> is powerful for all C<k>.
C<0> is returned for all negative or zero values of C<n>.

With C<k=2> this corresponds to Pari's C<ispowerful> function for positive
values of C<n>.  Pari chooses to define 0 as powerful and uses C<abs(n)>.

While we can easily code this as a one line function using
L</vecall> and L</factor_exp>, this is significantly faster and doesn't
need to fully factor the input.

=head2 powerful_numbers

  my $arrayref_pn1 = powerful_numbers(20);      # 1,4,8,9,16
  my $arrayref_pn2 = powerful_numbers(20,40);   # 25,27,32,36
  my $arrayref_pn3 = powerful_numbers(1,70,3);  # 1,8,16,27,32,64

Given a single non-negative integer C<n>, returns an array
reference with all 2-powerful integers from C<1> to C<n> inclusive.

Given two non-negative integers C<lo> and C<hi>, returns an array
reference with all 2-powerful integers from C<lo> to C<hi> inclusive.

Given three non-negative integers C<lo>, C<hi>, and C<k>, returns an array
reference with all C<k>-powerful integers from C<lo> to C<hi> inclusive.

  # Alternate solutions for values 1-n

  # Simple, but very slow for high $n.
  for (1..$n) { say if is_powerful($_,$k); }

  # Not so bad, especially for high $k.
  for (1..powerful_count($n,$k)) { say nth_powerful($_,$k); }

  # Best by far.
  say for @{powerful_numbers(1,$n,$k)};

Note that C<< n <= 0 >> are non-powerful.

=head2 powerful_count

  my $npowerful3 = powerful_count(2**32, 3);

Given an integer C<n> and an optional non-negative integer C<k>,
returns the total number of C<k>-powerful numbers
from C<1> to C<n> inclusive.
If C<k> is omitted, C<k=2> is used.

=head2 sumpowerful

Given an integer C<n> and an optional non-negative integer C<k>,
returns the sum of positive integer C<k>-powerful numbers less than or equal
to C<n>.
That is, the sum for all C<x>, C<< 1 <= x <= n >>,
where C<x> is a C<k>-powerful number.
If C<k> is omitted, C<k=2> is used.

=head2 nth_powerful

Given a non-negative integer C<n> and an optional non-negative integer C<k>,
returns the C<n>-th C<k>-powerful number.
If C<k> is omitted, C<k=2> is used.
For all C<k>, returns undef for C<n=0> and 1 for C<n=1>.


=head2 is_perfect_power

Given an integer C<n>, returns C<1> if C<n> is a perfect power,
and C<0> otherwise.  That is, if C<n = c^k> for some integers C<c>
and C<k> with C<k> greater than 1.

The results match the C<mpz_perfect_power_p(n)> function of GMP 4.0+.
Following GMP, SAGE, and FLINT, we treat -1, 0, and 1 as perfect powers.

For positive integers, this is L<OEIS series A001597|http://oeis.org/A001597>.

=head2 next_perfect_power

Given an integer C<n>, returns the smallest perfect power greater
than C<n>.  Similar in API to L</next_prime>, but returns the next
perfect power with exponent greater than 1.
Starting from C<0> this gives the sequence C<1,4,8,9,16,25,...>.

Negative inputs are supported, with the result being the nearest value
greater than C<n> where C<is_perfect_power> returns true.

=head2 prev_perfect_power

Given an integer C<n>, returns the greatest perfect power less than C<n>.
Similar in API to L</prev_prime>, but returns the previous perfect power
with exponent greater than 1.

Negative inputs are supported, with the result being the nearest value
less than C<n> where C<is_perfect_power> returns true.

=head2 perfect_power_count

Given a non-negative integer C<n>, returns the number of integers
not exceeding C<n> which are perfect powers.
If given two non-negative integers C<lo> and C<hi>, returns the count
of perfect powers between C<lo> and C<hi> inclusive.

By convention, numbers less than 1 are not counted.

This can be calculated extremely quickly (less than 100ns per call
for native size integers), so in most cases there is no need for the
approximations or bounds.

This is L<OEIS series A069623|http://oeis.org/A069623>.

=head2 perfect_power_count_approx

Given a non-negative integer C<n>, quickly returns a
good estimate of the count of perfect powers less than or equal to C<n>.

=head2 perfect_power_count_lower

Given a non-negative integer C<n>, quickly returns a
lower bound of the count of perfect powers less than or equal to C<n>.
The actual count will always be greater than or equal to the result.

=head2 perfect_power_count_upper

Given a non-negative integer C<n>, quickly returns an
upper bound of the count of perfect powers less than or equal to C<n>.
The actual count will always be less than or equal to the result.

=head2 nth_perfect_power

Given a non-negative integer C<n>, returns the C<n>-th perfect power.

Since the perfect power count can be calculated extremely quickly,
using inverse interpolation can calculate the C<n>-th perfect power
quite rapidly.

Similar to L</perfect_power_count>, the convention is to
exclude all integers less than 1.
Hence C<n=0> returns undef and C<n=1> returns 1.

=head2 nth_perfect_power_approx

Given a non-negative integer C<n>, quickly returns a
good estimate of the C<n>-th perfect power.

=head2 nth_perfect_power_lower

Given a non-negative integer C<n>, quickly returns a
lower bound of the C<n>-th perfect power.
The actual value will always be greater than or equal to the result.

=head2 nth_perfect_power_upper

Given a non-negative integer C<n>, quickly returns an
upper bound of the C<n>-th perfect power.
The actual value will always be less than or equal to the result.


=head2 next_chen_prime

Given a non-negative integer C<n>, return the smallest Chen prime
strictly greater than C<n>.
This will be a prime C<< p: p > n >>, where C<p+2> is either a
prime or a semiprime.


=head2 smooth_count

Given non-negative integer inputs C<n> and C<k>, returns the number of
integers between C<1> and C<n> inclusive, that have no prime factor larger
than C<k>.

For all C<n>, C<smooth_count(n,0) = smooth_count(n,1) = 1>.
For all C<k>, C<smooth_count(0,k) = 0> and C<smooth_count(1,k) = 1>.

This is equivalent to, but much faster than,
C<vecsum( map { is_smooth($_,$k) } 1..$n )>.

=head2 rough_count

Given non-negative integer inputs C<n> and C<k>, returns the number of
integers between C<1> and C<n> inclusive, that have no prime factor less
than C<k>.

For all C<n>, C<rough_count(n,0) = rough_count(n,1) = rough_count(n,2) = n>.
For all C<k>, C<rough_count(0,k) = 0> and C<rough_count(1,k) = 1>.

This is equivalent to, but much faster than,
C<vecsum( map { is_rough($_,$k) } 1..$n )>.

=head2 is_practical

Given an integer C<n>, returns 1 if C<n> is a practical number,
and returns 0 otherwise.
A practical number is a positive integer C<n> such that all smaller
positive integers can be represented as sums of distinct divisors of C<n>.
This is L<OEIS series A005153|http://oeis.org/A005153>.


=head2 carmichael_lambda

Given a non-negative integer C<n>, returns the Carmichael function
(also called the reduced totient function, or Carmichael λ(n)).
This is the smallest
positive integer C<m> such that C<a^m = 1 mod n> for every integer C<a>
coprime to C<n>.  This is L<OEIS series A002322|http://oeis.org/A002322>.

This corresponds to Mathematica's C<CarmichaelLambda[n]> function.
It can be computed in Pari/GP as C<lcm(znstar(n)[2])>.

=head2 kronecker

Given two integers C<a> and C<n>, returns the Kronecker symbol C<(a|n)>.
The possible return values with their meanings for odd prime C<n> are:

   0   a = 0 mod n
   1   a is a quadratic residue mod n       (x^2 = a mod n for some x)
  -1   a is a quadratic non-residue mod n   (no x where x^2 = a mod n)

The Kronecker symbol is an extension of the Jacobi symbol to all integer
values of C<n> from the latter's domain of positive odd values of C<n>.
The Jacobi symbol is itself an extension of the Legendre symbol, which is
only defined for odd prime values of C<n>.  This corresponds to Pari's
C<kronecker(a,n)> function, Mathematica's C<KroneckerSymbol[n,m]>
function, and GMP's C<mpz_kronecker(a,n)>, C<mpz_jacobi(a,n)>, and
C<mpz_legendre(a,n)> functions.

If C<n> is not an odd prime, then the result does not necessarily
indicate whether C<a> is a quadratic residue mod C<n>.  Using the function
L</is_qr> will return correct results for any C<n>, though could be slower.

=head2 factorial

Given a non-negative integer C<n>, returns the factorial of C<n>,
defined as the product of the integers 1 to C<n> with the special case
of C<factorial(0) = 1>.  This corresponds to Pari's C<factorial(n)>
and Mathematica's C<Factorial[n]> functions.

=head2 subfactorial

Given a non-negative integer C<n>, returns the subfactorial of C<n>,
which is the number of derangements of C<n> objects.  This is the number
of permutations of n items where each item is not allowed to stay in its
starting position.

This is L<OEIS series A000166|http://oeis.org/A000166>.
This corresponds to Mathematica's C<Subfactorial[n]> function.

=head2 binomial

Given two integers C<n> and C<k>, returns the binomial coefficient
C<n*(n-1)*...*(n-k+1)/k!>, also known as the choose function.  Negative
arguments use the L<Kronenburg extensions|http://arxiv.org/abs/1105.3689/>.
This corresponds to Pari's C<binomial(n,k)> function, Mathematica's
C<Binomial[n,k]> function, and GMP's C<mpz_bin_ui> function.

For negative arguments, this matches Mathematica.  Pari does not implement
the C<< n < 0, k <= n >> extension and instead returns C<0> for this
case.  GMP's API does not allow negative C<k> but otherwise matches.
C<Math::BigInt> version 1.999816 and later supports negative arguments
with similar semantics.  Prior to this, C<< n < 0, k > 0 >> was undefined.

=head2 falling_factorial

Given two integers C<x> and C<n>, with C<n> non-negative, returns the
falling factorial of C<n>.

  falling_factorial(x,n) = x * (x-1) * (x-2) * ... * (x-(n-1))

This corresponds to Mathematica's C<FactorialPower[x,n]> function.

=head2 rising_factorial

Given two integers C<x> and C<n>, with C<n> non-negative, returns the
rising factorial of C<n>.

  rising_factorial(x,n) = x * (x+1) * (x+2) * ... * (x+(n-1))

This corresponds to Mathematica's C<Pochhammer[x,n]> function.

=head2 powersum

  say powersum(100,1);  #     5050 = vecsum(1..100)
  say powersum(100,2);  #   338350 = vecsum(map{powint($_,2)} 1..100)
  say powersum(100,3);  # 25502500 = vecsum(map{powint($_,3)} 1..100)

Given two non-negative integers C<n> and C<k>, returns the sum of C<k>-th
powers of the first C<n> positive integers.

With C<k=2> this is (L<OEIS A000330|http://oeis.org/A000330>).
With C<k=3> this is (L<OEIS A000537|http://oeis.org/A000537>).
With C<k=4> this is (L<OEIS A000538|http://oeis.org/A000538>).
OEIS sequences can be found through C<k=8>.

This corresponds to the C<faulhaber_sum(n,k)> function in L<Math::AnyNum>
and Pari's C<dirpowerssum(n,k)> function using integer arguments.


=head2 hclassno

Given an integer C<n>, returns 12 times the
Hurwitz-Kronecker class number.
This will always be an integer due to the pre-multiplication by 12.
The result is C<0> for negative C<n> and all C<n> congruent to 1 or 2 mod 4.

This is related to Pari's C<qfbhclassno(n)> where C<hclassno(n)> for positive
C<n> equals C<12 * qfbhclassno(n)> in Pari/GP.
This is L<OEIS A259825|http://oeis.org/A259825>.


=head2 bernfrac

  my($num,$den) = bernfrac(12);  # returns (-691,2730)

Returns the Bernoulli number C<B_n> for an integer argument C<n>, as a
rational number represented by two integers.  B_1 is chosen as 1/2, which
is the same as Pari's C<bernfrac(n)> and Mathematica's C<BernoulliB>
functions.

Having a modern version of L<Math::Prime::Util::GMP> installed will make
a big difference in speed.  That module uses a fast Pi/Zeta method.
Our pure Perl backend uses the Seidel method as shown by Peter Luschny.
This is faster than L<Math::Pari> which uses an older algorithm,
but quite a bit slower than modern Pari, Mathematica, or our GMP backend.

This corresponds to Pari's C<bernfrac> function
and Mathematica's C<BernoulliB> function.

=head2 bernreal

Given a non-negative integer C<n>, returns the Bernoulli number C<B_n>
as a L<Math::BigFloat> object using the default precision.  An optional
second argument may be given specifying the precision to be used.

This corresponds to Pari's C<bernreal> function.

=head2 stirling

  say "s(14,2) = ", stirling(14, 2);
  say "S(14,2) = ", stirling(14, 2, 2);
  say "L(14,2) = ", stirling(14, 2, 3);

Given two 32-/64-bit non-negative integers C<n> and C<k>, plus an
optional third argument C<kind> (1, 2, or 3, with the default being 1),
returns the Stirling number of the given kind.
The third kind are the unsigned Lah numbers.
This corresponds to Pari's C<stirling(n,k,{type})>
function and Mathematica's C<StirlingS1> / C<StirlingS2> functions.

Stirling numbers of the first kind are C<-1^(n-k)> times the number of
permutations of C<n> symbols with exactly C<k> cycles.  Stirling numbers
of the second kind are the number of ways to partition a set of C<n>
elements into C<k> non-empty subsets.  The Lah numbers are the number of
ways to split a set of C<n> elements into C<k> non-empty lists.

=head2 fubini

Given a non-negative integer C<n>, returns the Fubini number of n,
also called the ordered Bell numbers, or the number of ordered partitions
of C<n>.  It is the count of rankings of C<n> items allowing for ties.

This is the L<OEIS series A000670|http://oeis.org/A000670>.

=head2 harmfrac

  my($num,$den) = harmfrac(12);  # returns (86021,27720)

Given a non-negative integer C<n>, returns the Harmonic number C<H_n> as a
rational number represented by two integers.  The harmonic
numbers are the sum of reciprocals of the first C<n> natural numbers:
C<1 + 1/2 + 1/3 + ... + 1/n>.

Binary splitting (Fredrik Johansson's elegant formulation) is used.

This corresponds to Mathematica's C<HarmonicNumber> function.

=head2 harmreal

Given a non-negative integer C<n>, returns the Harmonic number C<H_n> as a
L<Math::BigFloat> object using the default precision.  An optional
second integer argument may be given specifying the precision to be used.

For large C<n> values, using a lower precision may result in faster
computation as an asymptotic formula may be used.  For precisions of
13 or less, native floating point is used for even more speed.

=head2 legendre_phi

  $phi = legendre_phi(1000000000, 41);

Given two non-negative integers C<n> and C<a>, returns the Legendre phi
function (also called the Legendre sum).
This is the count of positive integers C<< <= n >> which are not
divisible by any of the first C<a> primes.

This corresponds to the C<legendre_phi(n,a)> function in SAGE, and the
C<--phi n a> feature of C<primecount>.

=head2 inverse_li

  $approx_prime_count = inverse_li(1000000000);

Given a non-negative integer C<n>, returns the least integer value C<k>
such that C<< Li(k) >= n >>.  Since the logarithmic integral C<Li(n)> is
a good approximation to the number of primes less than C<n>, this function
is a good simple approximation to the nth prime.

=head2 inverse_li_nv

  $faster_approx_prime_count = inverse_li_nv(1000000000);

With input C<x> and output both in NV (floating point), computes the
inverse of the logarithmic integral.  This should be very fast, as everything
is done in native long double precision, no Perl bigints or bigfloats are
involved, and the computed result is returned as an NV.

The L</inverse_li> function uses this to start, then ensures the integer
return value is the closest inverse of the integer result of the
L</LogarithmicIntegral> function.  While this is a small amount of extra
time for small inputs, once we have to go to Perl and use BigInt / BigFloat,
the extra time can be significant.


=head2 numtoperm

  @p = numtoperm(10,654321);  # @p=(1,8,2,7,6,5,3,4,9,0)

Given a non-negative integer C<n> and integer C<k>, return the
rank C<k> lexicographic permutation of C<n> elements.
C<k> will be interpreted as mod C<n!>.

This will match iteration number C<k> (zero based) of L</forperm>.

This corresponds to Pari's C<numtoperm(n,k)> function (Pari 2.6 and
later use the same lexicographic ordering).

=head2 permtonum

  $k = permtonum([1,8,2,7,6,5,3,4,9,0]);  # $k = 654321

Given an array reference containing each integer from C<0> to C<n-1>,
in some ordering, returns the lexicographic permutation rank of the set.
This is the inverse of the L</numtoperm> function.

This will match iteration number C<k> (zero based) of L</forperm>.
The result will be between C<0> and C<n!-1>.

This corresponds to Pari's C<permtonum(n)> function (Pari 2.6 and
later use the same lexicographic ordering).

=head2 randperm

  @p = randperm(100);   # returns shuffled 0..99
  @p = randperm(100,4)  # returns 4 elements from shuffled 0..99
  @s = @data[randperm(1+$#data)];    # shuffle an array
  @p = @data[randperm(1+$#data,2)];  # pick 2 from an array

Given a single non-negative integer C<n>, returns a random permutation
of the integers from C<0> to C<n-1>.

Optionally takes a second non-negative integer argument C<k>.
The returned list will then have only C<k> elements.
This is more efficient than truncating the full shuffled list.

The randomness comes from our CSPRNG.

The slicing technique shown in the last example is similar to L</vecsample>.

=head2 shuffle

  @shuffled = shuffle(@data);

Takes a list as input, and returns a random permutation of the list.
Like randperm, the randomness comes from our CSPRNG.

This function is functionally identical to the C<shuffle> function
in L<List::Util>.  The only difference is the random source (Chacha20
with better randomness, a larger period, and a larger state).  This
does make it slower.

If the entire shuffled array is desired, this is faster than slicing
with L</randperm> as shown in its example above.  If fewer elements
are needed (a "pick" or "sample") then L</vecsample> or slicing with
L</randperm> will be much more efficient.

=head2 vecsample

  $oneof = vecsample(1,@data);  # Select one random value
  @twoof = vecsample(2,@data);  # Select two random values

Takes a non-negative integer C<k> and a list, and returns C<k> randomly
selected values from the list.  The randomness comes from our CSPRNG.

If the input is exactly two elements (C<k> and one other) and the second
value is an array reference, then we will use it as the input list:

  $oneof = vecsample(1, $arrayref);
  @twoof = vecsample(1, \@data);

This can be a large performance increase if the input list is large
(e.g. 2x at 1000 elements, can be 10x with more).
While there might be confusion when sampling a list with exactly
one element, where that element is an array reference, this is
assumed to be a rare case.

This is similar to C<sample> from L<List::Util>, C<choose_multiple> from
Rust rand, and Raku's C<pick>.



=head1 MODULAR ARITHMETIC

=head2 OVERVIEW

Functions for fast modular arithmetic are provided:
add, subtract, multiply, divide, power, square root, nth root, inverse.
Additionally, fast modular calculation of factorial, binomial,
and Lucas sequences are provided.
See L</"MODULAR FUNCTIONS"> for more functions that operate mod n.

Semantics mostly follow Pari/GP, though in some cases they will indicate
an error while we return undef.

  We use the absolute value of the modulus.
  A modulus of zero returns undef.
  A modulus of 1 will return 0.
  If a modular result doesn't exist, we return undef.

=head2 negmod

Given two integers C<a> and C<n>, return C<-a mod |n|>.

This is similar to C<submod(0,$a,$n)> or C<$n ? modint(-$a,absint($n)) : undef>.

=head2 addmod

Given three integers C<a>, C<b>, and C<n>, return C<(a+b) mod |n|>.
This is particularly useful when dealing with numbers that are larger
than a half-word but still native size.
No bigint package is needed and this can be 10-200x faster than using one.

=head2 submod

Given three integers C<a>, C<b>, and C<n>, return C<(a-b) mod |n|>.

=head2 mulmod

Given three integers C<a>, C<b>, and C<n>, return C<(a*b) mod |n|>.
This is particularly useful when C<n> fits in a native integer.
No bigint package is needed and this can be 10-200x faster than using one.

=head2 muladdmod

Given four integers C<a>, C<b>, C<c>, and C<n>, return C<(a*b+c) mod |n|>.

=head2 mulsubmod

Given four integers C<a>, C<b>, C<c>, and C<n>, return C<(a*b-c) mod |n|>.

=head2 divmod

Given three integers C<a>, C<b>, and C<n>, return C<(a/b) mod |n|>.
This is done as C<(a * (1/b mod |n|)) mod |n|>.
If no inverse of C<b> mod C<|n|> exists then undef is returned.

=head2 powmod

Given three integers C<a>, C<b>, and C<n>, return C<(a ** b) mod |n|>.
Typically binary exponentiation is used, so the process is very efficient.
With native size inputs, no bigint library is needed.

C<powmod(a,-b,n)> is calculated as C<powmod(invmod(a,n),b,n)>.
If C<1/a mod |n|> does not exist, undef is returned.

=head2 sqrtmod

Given two integers C<a> and C<n>, return the square root of C<a> mod C<|n|>.
If no square root exists, undef is returned.  If defined, the return value
C<r> will always satisfy C<r^2 = a mod |n|>.

If the modulus is prime, the function will always return C<r>, the smaller
of the two square roots (the other being C<-r mod |n|>.  If the modulus is
composite, one of possibly many square roots will be returned, and it will
not necessarily be the smallest.

=head2 allsqrtmod

Given two integers C<a> and C<n>, returns a sorted list of all modular
square roots of C<a> mod C<|n|>. If no square root exists, an empty
list is returned.

Some inputs will return very many roots.
For example, C<a = p^4, n = 24 * p^4> for prime p, has many roots,
and C<sqrtmod(89**8, 24*89**8)> has over 500 million.

=head2 rootmod

Given three integers C<a>, C<k>, and C<n>, returns a C<k>-th root of
C<a> modulo C<|n|>, or undef if one does not exist.
If defined, the return value C<r> will satisfy C<r^k = a mod |n|>.
There is no guarantee that the smallest root will be returned.

For some composites with large prime powers this may not be efficient.

C<rootmod(a,-k,n)> is calculated as C<rootmod(invmod(a,n),k,n)>.
If C<1/a mod |n|> does not exist, undef is returned.

=head2 allrootmod

Given three integers C<a>, C<k>, and C<n>, returns a sorted list of all
modular C<k>-th root of C<a> modulo C<|n|>.
If no root exists, an empty list is returned.

Similar to L</allsqrtmod>, some inputs have millions or billions of roots,
so it might not be able to successfully return them all.

=head2 invmod

  say "The inverse of 42 mod 2017 = ", invmod(42,2017);

Given two integers C<a> and C<n>, return the inverse of C<a> modulo C<|n|>.
If not defined, undef is returned.  If defined, then the return value
multiplied by C<a> equals C<1> modulo C<|n|>.

The results correspond to the Pari result of C<lift(Mod(1/a,n))>.  The
semantics with respect to negative arguments match Pari.  Notably, a
negative C<n> is negated, which is different from Math::BigInt, but in both
cases the return value is still congruent to C<1> modulo C<n> as expected.

Mathematica uses C<Powermod[a, -1, n]>, where C<n> must be positive.

=head2 factorialmod

Given a non-negative integer C<n> and an integer C<m>, returns C<n! mod |m|>.
This is much faster than computing the large C<factorial(n)> followed
by a mod operation.

While very efficient, this is not state of the art.  Currently,
Fredrik Johansson's fast multi-point polynomial evaluation method as
used in FLINT is the fastest known implementation.
This becomes noticeable for C<< n > 10^8 >> or so,
and the O(n^.5) versus O(n) complexity is very apparent with large C<n>.

=head2 binomialmod

Given integer arguments C<n>, C<k>, and C<m>, returns C<binomial(n,k) mod |m|>.
This is much faster than computing the large C<binomial(n,k)> followed
by a mod operation.

C<|m|> does not need to be prime.
The result is extended to negative C<n>.
Negative C<k> will return zero.

This corresponds to Mathematica's C<BinomialMod[n,m,p]> function.  It has
similar functionality to Max Alekseyev's C<binomod.gp> Pari routine.

=head2 lucasumod

Given integers C<P>, C<Q>, the non-negative integer C<k>, and the
integer C<n>, efficiently compute C<lucasu(P,Q,k) mod |n|>.

This corresponds to gmpy2's C<lucasu_mod> function.

When C<(P,Q) = (1,-1)> this returns the modular Fibonacci sequence.  This
corresponds to Sidef's C<fibmod> function.

=head2 lucasvmod

Given integers C<P>, C<Q>, the non-negative integer C<k>, and the
integer C<n>, efficiently compute C<lucasv(P,Q,k) mod |n|>.

This corresponds to gmpy2's C<lucasv_mod> function.

=head2 lucasuvmod

  # Compute the 5000-th Fibonacci and Lucas numbers, mod 1001
  ($U,$V) = lucasuvmod(1, -1, 5000, 1001);

Given integers C<P>, C<Q>, the non-negative integer C<k>, and the
integer C<n>, efficiently compute the k-th value
of C<U(P,Q) mod |n|> and C<V(P,Q) mod |n|>.

This is similar to the L</lucas_sequence> function, but uses a more
consistent argument order and does not return C<Q_k>.

=head2 lucas_sequence

  my($U, $V, $Qk) = lucas_sequence($n, $P, $Q, $k);

B<lucas_sequence() is deprecated.  Use lucasuvmod() instead.>

Computes C<U_k>, C<V_k>, and C<Q_k> for the Lucas sequence defined by
C<P>,C<Q>, modulo C<|n|>.  The modular Lucas sequence is used in a
number of primality tests and proofs.
C<k> must be non-negative, and C<n> must be non-zero.

=head2 pisano_period

Given a non-negative integer C<n>, returns the period of the Fibonacci
sequence modulo C<n>.
The modular Fibonacci numbers can be produced using C<lucasumod(1,-1,k,n)>.
They are periodic for any integer C<n>, and the Pisano period is the
length of the repeating sequence.

This is the L<OEIS series A001175|http://oeis.org/A001175>.

=head1 MODULAR FUNCTIONS

=head2 OVERVIEW

More functions are provided that operate mod n.  They use similar semantics
with respect to the modulus: the absolute value is used, and a modulus of 0
will return undef.  However the behavior with C<n = 1> is not always the same.

=head2 znlog

  $k = znlog($a, $g, $p)

Returns the integer C<k> that solves the equation C<a = g^k mod |p|>, or
undef if no solution is found.  This is the discrete logarithm problem.

The implementation for native integers first applies Silver-Pohlig-Hellman
on the group order to possibly reduce the problem to a set of smaller
problems.  The solutions are then performed using a mixture of trial,
Shanks' BSGS, and Pollard's DLP Rho.

The PP implementation is less sophisticated, with only a memory-heavy BSGS
being used.

=head2 znorder

  $order = znorder(2, next_prime(10**16)-6);

Given two positive integers C<a> and C<n>, returns the multiplicative order
of C<a> modulo C<|n|>.  This is the smallest positive integer C<k> such that
C<a^k ≡ 1 mod |n|>.  Returns undef if C<n = 0>, C<a = 0>, or if
C<a> and C<n> are not coprime, since no value can result in 1 mod n.
Returns 1 if C<a = 1> or if C<n = 1>.

Note the latter differs from other mod functions, because the return value
is a positive integer, not an integer mod n.

This corresponds to Pari's C<znorder(Mod(a,n))> function and Mathematica's
C<MultiplicativeOrder[a,n]> function.

=head2 znprimroot

Given an integer C<n>, where C<n> is treated as C<|n|>,
returns the smallest primitive root of C<(Z/nZ)^*>,
or C<undef> if no root exists.
A root exists when C<euler_phi($n) == carmichael_lambda($n)>,
which will be true only if
C<< n is one of {2, 4, p^k, 2p^k} >> for odd prime p.

Like other modular functions, if C<n = 0> the function returns undef.

L<OEIS A033948|http://oeis.org/A033948> is a sequence of integers where
the primitive root exists, while L<OEIS A046145|http://oeis.org/A046145>
is a list of the smallest primitive roots, which is what this function
produces.

=head2 is_primitive_root

Given two integers C<a> and C<n>, returns C<1> if C<a> is a
primitive root modulo C<|n|>, and C<0> if not.  If C<a> is a primitive root,
then C<euler_phi(n)> is the smallest C<e> for which C<a^e = 1 mod n>.

Like other modular functions, if C<n = 0> the function returns undef.

=head2 qnr

Given an integer C<n>, returns the least quadratic non-residue
modulo C<|n|>.  This is the smallest integer C<a> where there does not
exist an integer C<b> such that C<a = b^2 mod |n|>.

Like other modular functions, if C<n = 0> the function returns undef.

This is L<OEIS A020649|http://oeis.org/A020649>.
For primes it is L<OEIS A053760|http://oeis.org/A053760>.

=head2 is_qr

Given two integers C<a> and C<n>, returns 1 if C<a> is a
quadratic residue modulo C<|n|>, and 0 otherwise.
A return value of 1 indicates there exists an C<x> where C<a = x^2 mod |n|>.

For odd primes, this is similar to checking C<a==0 || kronecker(a,n) == 1>.

For all values, this will be equal to C<sqrtmod(a,n) != undef>, with
possibly better performance.

Like other modular functions, if C<n = 0> the function returns undef.

=head1 RANDOM NUMBERS

=head2 OVERVIEW

Prior to version 5.20, Perl's C<rand> function used the system rand function.
This meant it varied by system, and was almost always a poor choice.  For
5.20, Perl standardized on C<drand48> and includes the source so there are no
system dependencies.  While this was an improvement, C<drand48> is not a good
PRNG.  It really only has 32 bits of random values, and fails many statistical
tests.  See L<http://www.pcg-random.org/statistical-tests.html> for more
information.

There are much better choices for standard random number generators, such as the
Mersenne Twister, PCG, or Xoroshiro128+.  Someday perhaps Perl will get one of
these to replace drand48.  In the mean time, L<Math::Random::MTwist> provides
numerous features and excellent performance, or this module.

Since we often deal with random primes for cryptographic purposes, we have
additional requirements.  This module uses a CSPRNG for its random stream.
In particular, ChaCha20, which is the same algorithm used by BSD's
C<arc4random> and C</dev/urandom> on BSD and Linux 4.8+.
Seeding is performed at startup using the Win32 Crypto API (on Windows),
C</dev/urandom>, C</dev/random>, or L<Crypt::PRNG>, whichever is found first.

We use the original ChaCha definition rather than RFC7539.  This means a
64-bit counter, resulting in a period of 2^72 bytes or 2^68 calls to
L</drand> or L</irand64>.
This compares favorably to the 2^48 period of Perl's C<drand48>.
It has a 512-bit state which is significantly larger than the
48-bit C<drand48> state.  When seeding, 320 bits (40 bytes) are used.
Among other things, this means all 52! permutations of a shuffled card deck
are possible, which is not true of L<List::Util/shuffle>.

One might think that performance would suffer from using a CSPRNG, but
benchmarking shows this does not seem to be the case.
The speed of C<irand>, C<irand64>, and C<drand> is within 20% of the fastest
existing modules using non-CSPRNG methods, and 2 to 20 times faster
than most.  While a faster underlying RNG is useful, the Perl call
interface overhead is a majority of the time for these calls.
Carefully tuning that interface is critical for any module.

For performance on large amounts of data, see the tables
in L</random_bytes>.

Each thread uses its own context, meaning seeding in one thread has no
impact on other threads.  In addition to improved security, this is
better for performance than a single context with locks.
If explicit control of multiple independent streams is needed then using
a more specific module is recommended.  I believe L<Crypt::PRNG>
(part of L<CryptX>) and L<Bytes::Random::Secure> are good alternatives.

Using the C<:rand> export option will define C<rand> and C<srand> as similar
but improved versions of the system functions of the same name, as well as
L</irand> and L</irand64>.


=head2 irand

  $n32 = irand;     # random 32-bit integer

Returns a random 32-bit integer using the CSPRNG.

=head2 irand64

  $n64 = irand64;   # random 64-bit integer

Returns a random 64-bit integer using the CSPRNG (on 64-bit Perl).
On a 32-bit Perl, it returns the maximum UV bits, which will be only 32.

=head2 drand

  $f = drand;       # random floating point value in [0,1)
  $r = drand(25.33);   # random floating point value in [0,25.33)

Returns a random NV (Perl's native floating point) using the CSPRNG.  The
API is similar to Perl's C<rand> but giving better results.

The number of actual random bits will be equal to the number of mantissa
bits in the NV type.  For IEEE-754 doubles, this means 53 bits, and can
go to 64 or 113 bits with long double / quadmath support.
The L</_nvmantbits> function allows seeing how many bits are used.

This gives I<substantially> better quality random numbers than the default Perl
C<rand> function.  Among other things, on modern Perl's, C<rand> uses drand48,
which has 32 bits of not-very-good randomness and 16 more bits of obvious
patterns (e.g. the 48th bit alternates, the 47th has a period of 4, etc.).
Output from C<rand> fails at least 5 tests from the TestU01 SmallCrush suite,
while our function easily passes.

With the ":rand" tag, this function is additionally exported as C<rand>.

=head2 random_bytes

  $str = random_bytes(32);     # 32 random bytes

Given an unsigned number C<n> of bytes, returns a string filled with random
data from the CSPRNG.  Performance for large quantities:

    Module/Method                  Rate   Type
    -------------             ---------   ----------------------

    Math::Prime::Util::GMP    1067 MB/s   CSPRNG - ISAAC
    ntheory random_bytes       384 MB/s   CSPRNG - ChaCha20
    Crypt::PRNG                140 MB/s   CSPRNG - Fortuna
    Crypt::OpenSSL::Random      32 MB/s   CSPRNG - SHA1 counter
    Math::Random::ISAAC::XS     15 MB/s   CSPRNG - ISAAC
    ntheory entropy_bytes       13 MB/s   CSPRNG - /dev/urandom
    Crypt::Random               12 MB/s   CSPRNG - /dev/urandom
    Crypt::Urandom              12 MB/s   CSPRNG - /dev/urandom
    Bytes::Random::Secure        6 MB/s   CSPRNG - ISAAC
    ntheory pure perl ISAAC      5 MB/s   CSPRNG - ISAAC (no XS)
    Math::Random::ISAAC::PP      2.5 MB/s CSPRNG - ISAAC (no XS)
    ntheory pure perl ChaCha     1.0 MB/s CSPRNG - ChaCha20 (no XS)
    Data::Entropy::Algorithms    0.5 MB/s CSPRNG - AES-CTR

    Math::Random::MTwist       927 MB/s   PRNG - Mersenne Twister
    Bytes::Random::XS          109 MB/s   PRNG - drand48
    pack CORE::rand             25 MB/s   PRNG - drand48 (no XS)
    Bytes::Random                2.6 MB/s PRNG - drand48 (no XS)

=head2 entropy_bytes

Similar to random_bytes, but directly using the entropy source.
This is not normally recommended as it can consume shared system
resources and is typically slow -- on the computer that produced
the L</random_bytes> chart above, using C<dd> generated the same
13 MB/s performance as our L</entropy_bytes> function.

The actual performance will be highly system dependent.

=head2 urandomb

  $n32 = urandomb(32);    # Classic irand32, returns a UV
  $n   = urandomb(1024);  # Random integer less than 2^1024

Given a number of bits C<b>, returns a random unsigned integer
less than C<2^b>.  The result will be uniformly distributed
between C<0> and C<2^b-1> inclusive.

=head2 urandomm

  $n = urandomm(100);    # random integer in [0,99]
  $n = urandomm(1024);   # random integer in [0,1023]

Given a positive integer C<n>, returns a random unsigned integer
less than C<n>.  The results will be uniformly distributed between
C<0> and C<n-1> inclusive.  Care is taken to prevent modulo bias.

=head2 csrand

Takes a binary string C<data> as input and seeds the internal CSPRNG.
This is not normally needed as system entropy is used as a seed on
startup.  For best security this should be 16-128 bytes of good
entropy.  No more than 1024 bytes will be used.

With no argument, reseeds using system entropy, which is preferred.

If the C<secure> configuration has been set, then this will croak if
given an argument.  This allows for control of reseeding with entropy
the module gets itself, but not user supplied.

=head2 srand

Takes a single UV argument and seeds the CSPRNG with it, as well as
returning it.  If no argument is given, a new UV seed is constructed.
Note that this creates a very weak seed from a cryptographic
standpoint, so it is useful for testing or simulations but
L</csrand> is recommended, or keep using the system entropy default seed.

The API is nearly identical to the system function C<srand>.  It
uses a UV which can be 64-bit rather than always 32-bit.  The
behaviour for C<undef>, empty string, empty list, etc. is slightly
different (we treat these as 0).

This function is not exported with the ":all" tag, but is with ":rand".

If the C<secure> configuration has been set, this function will croak.
Manual seeding using C<srand> is not compatible with cryptographic security.

=head2 rand

An alias for L</drand>, not exported unless the ":rand" tag is used.

=head2 random_factored_integer

  my($n, $factors) = random_factored_integer(1000000);

Given a positive non-zero input C<n>, returns a uniform random integer
in the range C<1> to C<n>, along with an array reference containing
the factors.

This uses Kalai's algorithm for generating random integers along with
their factorization, and is much faster than the naive method of
generating random integers followed by a factorization.
A later implementation may use Bach's more efficient algorithm.


=head1 RANDOM PRIMES

=head2 random_prime

  my $small_prime = random_prime(1000);      # random prime <= limit
  my $rand_prime = random_prime(100, 10000); # random prime within a range

Returns a pseudo-randomly selected prime that will be greater than or equal
to the lower limit and less than or equal to the upper limit.  If no lower
limit is given, 2 is implied.  Returns undef if no primes exist within the
range.

The goal is to return a uniform distribution of the primes in the range,
meaning for each prime in the range, the chances are equally likely that it
will be seen.  This is removes from consideration such algorithms as
C<PRIMEINC>, which although efficient, gives very non-random output.  This
also implies that the numbers will not be evenly distributed, since the
primes are not evenly distributed.  Stated differently, the random prime
functions return a uniformly selected prime from the set of primes within
the range.  Hence given C<random_prime(1000)>, the numbers 2, 3, 487, 631,
and 997 all have the same probability of being returned.

For small numbers, a random index selection is done, which gives ideal
uniformity and is very efficient with small inputs.  For ranges larger than
this ~16-bit threshold but within the native bit size, a Monte Carlo method
is used.  This also
gives ideal uniformity and can be very fast for reasonably sized ranges.
For even larger numbers, we partition the range, choose a random partition,
then select a random prime from the partition.  This gives some loss of
uniformity but results in many fewer bits of randomness being consumed as
well as being much faster.


=head2 random_ndigit_prime

  say "My 4-digit prime number is: ", random_ndigit_prime(4);

Selects a random n-digit prime, where the input is an integer number of
digits.  One of the primes within that range (e.g. 1000 - 9999 for
4-digits) will be uniformly selected.

If the number of digits is greater than or equal to the maximum native type,
then the result will be returned as a BigInt.  However, if the C<nobigint>
configuration option is on, then output will be restricted to native size
numbers, and requests for more digits than natively supported will result
in an error.
For better performance with large bit sizes, install L<Math::Prime::Util::GMP>.


=head2 random_nbit_prime

  my $bigprime = random_nbit_prime(512);

Selects a random n-bit prime, where the input is an integer number of bits.
A prime with the nth bit set will be uniformly selected.

For bit sizes of 64 and lower, L</random_prime> is used, which gives completely
uniform results in this range.  For sizes larger than 64, Algorithm 1 of
Fouque and Tibouchi (2011) is used, wherein we select a random odd number
for the lower bits, then loop selecting random upper bits until the result
is prime.  This allows a more uniform distribution than the general
L</random_prime> case while running slightly faster (in contrast, for large
bit sizes L</random_prime> selects a random upper partition then loops
on the values within the partition, which very slightly skews the results
towards smaller numbers).

The result will be a BigInt if the number of bits is greater than the native
bit size.  For better performance with large bit sizes, install
L<Math::Prime::Util::GMP>.


=head2 random_safe_prime

  my $bigprime = random_safe_prime(512);

Produces an n-bit safe prime.  This is a prime C<p> where C<p = 2q+1> and
C<q> is also prime.

These types of primes are sometimes useful for discrete logarithm based
cryptography, and can be generated more efficiently using
simultaneous sieving.


=head2 random_strong_prime

  my $bigprime = random_strong_prime(512);

Constructs an n-bit strong prime using Gordon's algorithm.  We consider a
strong prime I<p> to be one where

=over

=item * I<p> is large.   This function requires at least 128 bits.

=item * I<p-1> has a large prime factor I<r>.

=item * I<p+1> has a large prime factor I<s>

=item * I<r-1> has a large prime factor I<t>

=back

Using a strong prime in cryptography guards against easy factoring with
algorithms like Pollard's Rho.  Rivest and Silverman (1999) present a case
that using strong primes is unnecessary, and most modern cryptographic systems
agree.  First, the smoothness does not affect more modern factoring methods
such as ECM.  Second, modern factoring methods like GNFS are far faster than
either method so makes the point moot.  Third, due to key size growth and
advances in factoring and attacks, for practical purposes, using large random
primes offers security equivalent to strong primes.

Similar to L</random_nbit_prime>, the result will be a BigInt if the
number of bits is greater than the native bit size.  For better performance
with large bit sizes, install L<Math::Prime::Util::GMP>.


=head2 random_proven_prime

  my $bigprime = random_proven_prime(512);

Constructs an n-bit random proven prime.  Internally this may use
L</is_provable_prime>(L</random_nbit_prime>) or
L</random_maurer_prime> depending on the platform and bit size.


=head2 random_proven_prime_with_cert

  my($n, $cert) = random_proven_prime_with_cert(512);

Similar to L</random_proven_prime>, but returns a two-element array containing
the n-bit provable prime along with a primality certificate.  The certificate
is the same as produced by L</prime_certificate> or
L</is_provable_prime_with_cert>, and can be parsed by L</verify_prime> or
any other software that understands MPU primality certificates.


=head2 random_maurer_prime

  my $bigprime = random_maurer_prime(512);

Construct an n-bit provable prime, using the FastPrime algorithm of
Ueli Maurer (1995).  This is the same algorithm used by L<Crypt::Primes>.
Similar to L</random_nbit_prime>, the result will be a BigInt if the
number of bits is greater than the native bit size.

The performance with L<Math::Prime::Util::GMP> installed is hundreds
of times faster, so it is highly recommended.

The differences between this function and that in L<Crypt::Primes> are
described in the L</"SEE ALSO"> section.

Internally this additionally runs the BPSW probable prime test on every
partial result, and constructs a primality certificate for the final
result, which is verified.  These provide additional checks that the resulting
value has been properly constructed.

If you don't need absolutely proven results, then it is somewhat faster
to use L</random_nbit_prime> either by itself or with some additional tests,
e.g.  L</miller_rabin_random> and/or L</is_frobenius_underwood_pseudoprime>.
One could also run L</is_provable_prime> on the result, but this will be slow.

=head2 random_maurer_prime_with_cert

  my($n, $cert) = random_maurer_prime_with_cert(512);

As with L</random_maurer_prime>, but returns a two-element array containing
the n-bit provable prime along with a primality certificate.  The certificate
is the same as produced by L</prime_certificate> or
L</is_provable_prime_with_cert>, and can be parsed by L</verify_prime> or
any other software that understands MPU primality certificates.
The proof construction consists of a single chain of C<BLS3> types.


=head2 random_shawe_taylor_prime

  my $bigprime = random_shawe_taylor_prime(8192);

Construct an n-bit provable prime, using the Shawe-Taylor algorithm in
section C.6 of FIPS 186-4.  This uses 512 bits of randomness and SHA-256
as the hash.  This is a slightly simpler and older (1986) method than
Maurer's 1995 construction.  It is a bit faster than Maurer's method, and
uses less system entropy for large sizes.  The primary reason to use this
rather than Maurer's method is to use the FIPS 186-4 algorithm.

Similar to L</random_nbit_prime>, the result will be a BigInt if the
number of bits is greater than the native bit size.  For better performance
with large bit sizes, install L<Math::Prime::Util::GMP>.  Also see
L</random_maurer_prime> and L</random_proven_prime>.

Internally this additionally runs the BPSW probable prime test on every
partial result, and constructs a primality certificate for the final
result, which is verified.  These provide additional checks that the resulting
value has been properly constructed.


=head2 random_shawe_taylor_prime_with_cert

  my($n, $cert) = random_shawe_taylor_prime_with_cert(4096);

As with L</random_shawe_taylor_prime>, but returns a two-element array
containing the n-bit provable prime along with a primality certificate.
The certificate is the same as produced by L</prime_certificate> or
L</is_provable_prime_with_cert>, and can be parsed by L</verify_prime> or
any other software that understands MPU primality certificates.
The proof construction consists of a single chain of C<Pocklington> types.


=head2 random_semiprime

Takes a positive integer number of bits C<bits>, returns a
random semiprime of exactly C<bits> bits.
The result has exactly two prime factors (hence semiprime).

The factors will be approximately equal size, which is typical
for cryptographic use.  For example, a 64-bit semiprime of this
type is the product of two 32-bit primes.
C<bits> must be C<4> or greater.

Some effort is taken to select uniformly from the universe of
C<bits>-bit semiprimes.  This takes slightly longer than some
methods that do not select uniformly.

=head2 random_unrestricted_semiprime

Takes a positive integer number of bits C<bits>, returns a
random semiprime of exactly C<bits> bits.
The result has exactly two prime factors (hence semiprime).

The factors are uniformly selected from the universe of all
C<bits>-bit semiprimes.  This means semiprimes with one factor
equal to C<2> will be most common, C<3> next most common, etc.
C<bits> must be C<3> or greater.

Some effort is taken to select uniformly from the universe of
C<bits>-bit semiprimes.  This takes slightly longer than some
methods that do not select uniformly.


=head1 UTILITY FUNCTIONS

=head2 prime_precalc

  prime_precalc( 1_000_000_000 );

Let the module prepare for fast operation up to a specific number.  It is not
necessary to call this, but it gives you more control over when memory is
allocated and gives faster results for multiple calls in some cases.  In the
current implementation this will calculate a sieve for all numbers up to the
specified number.


=head2 prime_memfree

  prime_memfree;

Frees any extra memory the module may have allocated.  Like with
C<prime_precalc>, it is not necessary to call this, but if you're done
making calls, or want things cleaned up, you can use this.  The object method
might be a better choice for complicated uses.

=head2 Math::Prime::Util::MemFree->new

  use Math::Prime::Util::MemFree;
  my $mf = Math::Prime::Util::MemFree->new;
  # perform operations.  When $mf goes out of scope, memory will be recovered.

This is a more robust way of making sure any cached memory is freed, as it
will be handled by the last C<MemFree> object leaving scope.  This means if
your routines were inside an eval that died, things will still get cleaned up.
If you call another function that uses a MemFree object, the cache will stay
in place because you still have an object.


=head2 prime_get_config

  my $cached_up_to = prime_get_config->{'precalc_to'};

  # Print all configuration
  my $r=prime_get_config();
  say "$_ $r->{$_}" for sort (keys %$r);

Returns a reference to a hash of the current settings.  The hash is a copy of
the configuration, so changing it has no effect.  The settings include:

  verbose         verbose level.  1 or more will result in extra output.
  bigintclass     the bigint type name (default Math::BigInt)
  precalc_to      primes up to this number are calculated
  maxbits         the maximum number of bits for native operations
  xs              0 or 1, indicating the XS code is available
  gmp             0 or 1, indicating GMP code is available
  maxparam        the largest value for most functions, without bigint
  maxdigits       the max digits in a number, without bigint
  maxprime        the largest representable prime, without bigint
  maxprimeidx     the index of maxprime, without bigint
  assume_rh       whether to assume the Riemann hypothesis (default 0)
  secure          disable ability to manually seed the CSPRNG

=head2 prime_set_config

  prime_set_config( assume_rh => 1 );

  prime_set_config(bigint=>Math::GMPz);

Allows setting of some parameters.  Currently the only parameters are:

  verbose      The default setting of 0 will generate no extra output.
               Setting to 1 or higher results in extra output.  For
               example, at setting 1 the AKS algorithm will indicate
               the chosen r and s values.  At setting 2 it will output
               a sequence of dots indicating progress.  Similarly, for
               random_maurer_prime, setting 3 shows real time progress.
               Factoring large numbers is another place where verbose
               settings can give progress indications.

  bigint       You can give either a single object (e.g. a value of the
               class you want), or a comma separated list of class names.
               The first class we can load will be used for all operations
               that use a bigint.
               A warning will be produced if one was not found.

  trybigint    Exactly the same behavior as C<bigint> but no warning
               will be output if we couldn't load anything from the list.

  xs           Allows turning off the XS code, forcing the Pure Perl
               code to be used.  Set to 0 to disable XS, set to 1 to
               re-enable.  You probably will never want to do this.

  gmp          Allows turning off the use of L<Math::Prime::Util::GMP>,
               which means using Pure Perl code for big numbers.  Set
               to 0 to disable GMP, set to 1 to re-enable.
               You probably will never want to do this.

  assume_rh    Allows functions to assume the Riemann hypothesis is
               true if set to 1.  This defaults to 0.  Currently this
               setting only impacts prime count lower and upper
               bounds, but could later be applied to other areas such
               as primality testing.  A later version may also have a
               way to indicate whether no RH, RH, GRH, or ERH is to
               be assumed.

  secure       The CSPRNG may no longer be manually seeded.  Once set,
               this option cannot be disabled.  L</srand> will croak
               if called, and L</csrand> will croak if called with any
               arguments.  L</csrand> with no arguments is still allowed,
               as that will use system entropy without giving anything
               to the caller.  The point of this option is to ensure that
               any called functions do not try to control the RNG.


=head1 FACTORING FUNCTIONS

=head2 factor

  my @factors = factor(3_369_738_766_071_892_021);
  # returns (204518747,16476429743)

Produces the prime factors of a positive number input, in numerical order.
The product of the returned factors will be equal to the input.  C<n = 1>
will return an empty list, and C<n = 0> will return 0.  This matches Pari.

In scalar context, returns Ω(n), the total number of prime factors
(L<OEIS A001222|http://oeis.org/A001222>).
This corresponds to Pari's C<bigomega(n)> function and Mathematica's
C<PrimeOmega[n]> function.
This is the same result that we would get if we evaluated the resulting
array in scalar context.

The current algorithm does a little trial division, a check for perfect
powers, followed by combinations of Pollard's Rho, SQUFOF, and Pollard's
p-1.  The combination is applied to each non-prime factor found.

Factoring bigints works with pure Perl, and can be very handy on 32-bit
machines for numbers just over the 32-bit limit, but it can be B<very> slow
for "hard" numbers.  Installing the L<Math::Prime::Util::GMP> module will
speed up bigint factoring a B<lot>, and all future effort on large number
factoring will be in that module.  If you do not have that module for
some reason, use the GMP or Pari version of bigint if possible
(e.g. C<< use bigint try => 'GMP,Pari' >>), which will run 2-3x faster
(though still 100x slower than the real GMP code).


=head2 factor_exp

  my @factor_exponent_pairs = factor_exp(29513484000);
  # returns ([2,5], [3,4], [5,3], [7,2], [11,1], [13,2])
  # factor(29513484000)
  # returns (2,2,2,2,2,3,3,3,3,5,5,5,7,7,11,13,13)

Produces pairs of prime factors and exponents in numerical factor order.
This is more convenient for some algorithms.  This is the same form that
Mathematica's C<FactorInteger[n]> and Pari/GP's C<factorint> functions
return.  Note that L<Math::Pari> transposes the Pari result matrix.

In scalar context, returns ω(n), the number of unique prime factors
(L<OEIS A001221|http://oeis.org/A001221>).
This corresponds to Pari's C<omega(n)> function and Mathematica's
C<PrimeNu[n]> function.
This is the same result that we would get if we evaluated the resulting
array in scalar context.

The internals are identical to L</factor>, so all comments there apply.
Just the way the factors are arranged is different.


=head2 divisors

  my @divisors = divisors(30);   # returns (1, 2, 3, 5, 6, 10, 15, 30)

Produces all the divisors of a positive number input, including 1 and
the input number.  The divisors are a power set of multiplications of
the prime factors, returned as a uniqued sorted list.  The result is
identical to that of Pari's C<divisors> and Mathematica's C<Divisors[n]>
functions.

In scalar context this returns the sigma0 function
(see Hardy and Wright section 16.7).
This is L<OEIS A000005|http://oeis.org/A000005>.
The result is identical to evaluating the array in scalar context, but
more efficient.
This corresponds to Pari's C<numdiv> and Mathematica's
C<DivisorSigma[0,n]> functions.

Also see the L</fordivisors> function for looping over the divisors.

When C<n=0> we return the empty set (zero in scalar context).

An optional second positive integer argument C<k> indicates that the results
should not include any value larger than C<k>.  This is especially useful
when the number has thousands of divisors and we may only be interested in
the small ones.


=head2 trial_factor

  my @factors = trial_factor($n);

Produces the prime factors of a positive number input using trial division.
The factors will be in numerical order.
For large inputs this will be very slow.

An optional second argument will indicate an upper limit for factors.
Factors C<2>, C<3>, and C<5> are always pulled out.
Factors larger than the second argument will not be found and hence the last
value in the list might be composite.

Like all the specific-algorithm C<*_factor> routines, this is not exported
unless explicitly requested.

=head2 fermat_factor

  my @factors = fermat_factor($n);

Produces factors, not necessarily prime, of the positive number input.  The
particular algorithm is Knuth's algorithm C.  For small inputs this will be
very fast, but it slows down quite rapidly as the number of digits increases.
It is very fast for inputs with a factor close to the midpoint
(e.g. a semiprime p*q where p and q are the same number of digits).

=head2 holf_factor

  my @factors = holf_factor($n);

Produces factors, not necessarily prime, of the positive number input.  An
optional number of rounds can be given as a second parameter.  It is possible
the function will be unable to find a factor, in which case a single element,
the input, is returned.  This uses Hart's One Line Factorization with no
premultiplier.  It is an interesting alternative to Fermat's algorithm,
and there are some inputs it can rapidly factor.  Overall it has the
same advantages and disadvantages as Fermat's method.

=head2 lehman_factor

  my @factors = lehman_factor($n);

Produces factors, not necessarily prime, of the positive number input.  An
optional argument, defaulting to 0 (false), indicates whether to run trial
division.  Without trial division, is possible the function will be unable
to find a factor, in which case a single element, the input, is returned.

This is Warren D. Smith's Lehman core with minor modifications.  It is
limited to 42-bit inputs: C<< n < 8796393022208 >>.

=head2 squfof_factor

  my @factors = squfof_factor($n);

Produces factors, not necessarily prime, of the positive number input.  An
optional number of rounds can be given as a second parameter.  It is possible
the function will be unable to find a factor, in which case a single element,
the input, is returned.  This function typically runs very fast.

=head2 prho_factor

Pollard's rho factoring algorithm.  See L</pbrent_factor> for the shared
description of both functions.

=head2 pbrent_factor

  my @factors = prho_factor($n);
  my @factors = pbrent_factor($n);

  # Use a very small number of rounds
  my @factors = prho_factor($n, 1000);

Produces factors, not necessarily prime, of the positive number input.  An
optional number of rounds can be given as a second parameter.  These attempt
to find a single factor using Pollard's Rho algorithm, either the original
version or Brent's modified version.  These are more specialized algorithms
usually used for pre-factoring very large inputs, as they are very fast at
finding small factors.


=head2 pminus1_factor

  my @factors = pminus1_factor($n);
  my @factors = pminus1_factor($n, 1_000);          # set B1 smoothness
  my @factors = pminus1_factor($n, 1_000, 50_000);  # set B1 and B2

Produces factors, not necessarily prime, of the positive number input.  This
is Pollard's C<p-1> method, using two stages with default smoothness
settings of 1_000_000 for B1, and C<10 * B1> for B2.  This method can rapidly
find a factor C<p> of C<n> where C<p-1> is smooth (it has no large factors).

=head2 pplus1_factor

  my @factors = pplus1_factor($n);
  my @factors = pplus1_factor($n, 1_000);          # set B1 smoothness

Produces factors, not necessarily prime, of the positive number input.  This
is Williams' C<p+1> method, using one stage and two predefined initial points.

=head2 cheb_factor

  my @factors = cheb_factor($n);
  my @factors = cheb_factor($n, 1_000);          # set B1 smoothness

Produces factors, not necessarily prime, of the positive number input.
This uses the properties of Chebyshev polynomials
(particularly that C<T_mn(x) = T_m(T_n(x))>)
and their relationship with the Lucas sequence,
to find factors if C<p-1> or C<p+1> is smooth.

This generally works better than our L</pplus1_factor>, but is slower than our
L</pminus1_factor>.

=head2 ecm_factor

  my @factors = ecm_factor($n);
  my @factors = ecm_factor($n, 100, 400, 10);      # B1, B2, # of curves

Produces factors, not necessarily prime, of the positive number input.  This
is the elliptic curve method using two stages.



=head1 MATHEMATICAL FUNCTIONS

=head2 ExponentialIntegral

  my $Ei = ExponentialIntegral($x);

Given a non-zero floating point input C<x>, this returns the real-valued
exponential integral of C<x>, defined as the integral of C<e^t/t dt>
from C<-infinity> to C<x>.

For non-BigFloat inputs, the result should be accurate to at least 14
digits.

For BigFloat inputs, full accuracy and performance is obtained
only if L<Math::Prime::Util::GMP> is installed.
If this module is not available, then other methods are used and give
at least 14 digits of accuracy:
continued fractions (C<< x < -1 >>),
rational Chebyshev approximation (C<< -1 < x < 0 >>),
a convergent series (small positive C<x>),
or an asymptotic divergent series (large positive C<x>).
The accuracy() setting of the input is used to determine the output accuracy.

=head2 LogarithmicIntegral

  my $li = LogarithmicIntegral($x);

Given a non-negative floating point input, returns the floating point
logarithmic integral of C<x>, defined as the integral of C<dt/ln t>
from C<0> to C<x>.
If given a negative input, the function will croak.
The function returns 0 at C<x = 0>, and C<-infinity> at C<x = 1>.

This is often known as C<li(x)>.  A related function is the offset logarithmic
integral, sometimes known as C<Li(x)> which avoids the singularity at 1.  It
may be defined as C<Li(x) = li(x) - li(2)>.  Crandall and Pomerance use the
term C<li0> for this function, and define C<li(x) = Li0(x) - li0(2)>.  Due to
this terminology confusion, it is important to check which exact definition is
being used.

For non-BigFloat objects, the result should be accurate to at least 14
digits.

For BigFloat inputs, full accuracy and performance is obtained
only if L<Math::Prime::Util::GMP> is installed.
The accuracy() setting of the input is used to determine the output accuracy.

=head2 RiemannZeta

  my $z = RiemannZeta($s);

Given a non-negative floating point input C<s>, returns the floating
point value of ζ(s)-1, where ζ(s) is the Riemann zeta function.  One is
subtracted to ensure maximum precision for large values of C<s>.  The zeta
function is the sum from k=1 to infinity of C<1 / k^s>.  This function only
uses real arguments, so is more properly the Euler Zeta function.

For non-BigFloat objects, the result should be accurate to at least 14
digits.  The XS code uses a rational Chebyshev approximation between 0.5 and 5,
and a series for other values.  The PP code uses an identical series for all
values.

For BigFloat inputs, full accuracy and performance is obtained
only if L<Math::Prime::Util::GMP> is installed.
If this module is not available, then other methods are used and give
at least 14 digits of accuracy:
Either Borwein (1991) algorithm 2, or the basic series.
Math::BigFloat L<RT 43692|https://rt.cpan.org/Ticket/Display.html?id=43692>
can produce incorrect high-accuracy computations when GMP is not used.
The accuracy() setting of the input is used to determine the output accuracy.

=head2 RiemannR

  my $r = RiemannR($x);

Given a positive non-zero floating point input, returns the floating
point value of Riemann's R function.  Riemann's R function gives a very close
approximation to the prime counting function.

For non-BigFloat objects, the result should be accurate to at least 14
digits.

For BigFloat inputs, full accuracy and performance is obtained
only if L<Math::Prime::Util::GMP> is installed.
If that module is not available, accuracy should be 35 digits.
The accuracy() setting of the input is used to determine the output accuracy.

=head2 LambertW

Returns the principal branch of the Lambert W function of a real value.
Given a value C<k> this solves for C<W> in the equation C<k = We^W>.  The
input must not be less than C<-1/e>.  This corresponds to Pari's C<lambertw>
function and Mathematica's C<ProductLog> / C<LambertW> function.

This function handles all real value inputs with non-complex return values
from the principal branch.
Pari/GP's C<lambertw> prior to 2.15 (2022) was a subset of this.
Recent Pari/GP and Mathematica both have more complete functions with
both branches, and support for complex arguments and results.

Calculation will be done with C long doubles if the input is a standard
scalar, but if the input is a BigFloat type, then
extended precision results will be generated.
The accuracy() setting of the input is used to determine the output accuracy.

Speed of the native code is about half of the fastest native code
(Veberic's C++), and about 10x faster than Pari/GP.  However the bignum
calculation is slower than Pari/GP.

=head2 Pi

  my $tau = 2 * Pi;     # $tau = 6.28318530717959
  my $tau = 2 * Pi(40); # $tau = 6.283185307179586476925286766559005768394

With no arguments, returns the value of Pi as an NV.  With a positive
integer argument, returns the value of Pi with the requested number of
digits (including the leading 3).  The return value will be an NV if the
number of digits fits in an NV (typically 15 or less), or a L<Math::BigFloat>
object otherwise.

For sizes over 10k digits, having either
L<Math::Prime::Util::GMP> or L<Math::BigInt::GMP> installed will help
performance.  For sizes over 50k, GMP is highly recommended.


=head1 PLATFORM INTROSPECTION

=head2 OVERVIEW

We include a number of non-exported functions that are useful for internal
use but can also be useful for users.  These functions are subject to
change or deletion in future revisions.

=head2 _uvsize

Returns the size of a UV in bytes (typically 4 or 8).
The size of the basic integer type used in Perl and the C library.

=head2 _uvbits

Returns the size of a UV in bits (typically 32 or 64).

=head2 _ivsize

Returns the size of an IV in bytes (typically 4 or 8).
This is going to be the same as L</_uvsize>.

=head2 _nvsize

Returns the size of an NV in bytes (typically 4, 8, or 16).
It's quite possible other sizes could be seen on non-standard
configurations.  Usually we won't care about this directly.

=head2 _nvmantbits

Returns the size of the mantissa of Perl's NV floating point type, in bits.
This can vary widely, with C<23, 52, 112> all possible from mainstream
platforms and other numbers possible.

This gives the actual mantissa bits, not counting the implicit 1.  The
significand precision is therefore one higher than the value returned
by this function.  A typical IEEE-754 double will report 52 here, which
means integers up to C<2^53-1> are able to be accurately stored.

Perl prior to 5.23 did not configure this at build time.  We will guess
based on the byte size of the NV on an IEEE-754 machine.

=head2 _nvmantdigits

How many full decimal integer digits able to be stored in an NV.


=head1 EXAMPLES

Print Fibonacci numbers:

    perl -Mntheory=:all -E 'say lucasu(1,-1,$_) for 0..20'

Print strong pseudoprimes to base 17 up to 10M:

    # Similar to A001262's isStrongPsp function, but much faster
    perl -MMath::Prime::Util=:all -E 'foroddcomposites { say if is_strong_pseudoprime($_,17) } 10000000;'

Print some primes above 64-bit range:

    perl -MMath::Prime::Util=:all -Mbigint -E 'my $start=100000000000000000000; say join "\n", @{primes($start,$start+1000)}'

    # Another way
    perl -MMath::Prime::Util=:all -E 'forprimes { say } "100000000000000000039", "100000000000000000993"'

    # Similar using Math::Pari:
    # perl -MMath::Pari=:int,PARI,nextprime -E 'my $start = PARI "100000000000000000000"; my $end = $start+1000; my $p=nextprime($start); while ($p <= $end) { say $p; $p = nextprime($p+1); }'

Generate Carmichael numbers (L<OEIS A002997|http://oeis.org/A002997>):

    perl -Mntheory=:all -E 'foroddcomposites { say if is_carmichael($_) } 1e6;'

    # Less efficient, similar to Mathematica or MAGMA:
    perl -Mntheory=:all -E 'foroddcomposites { say if $_ % carmichael_lambda($_) == 1 } 1e6;'

Examining the η3(x) function of Planat and Solé (2011):

  sub nu3 {
    my $n = shift;
    my $phix = chebyshev_psi($n);
    my $nu3 = 0;
    foreach my $nu (1..3) {
      $nu3 += (moebius($nu)/$nu)*LogarithmicIntegral($phix**(1/$nu));
    }
    return $nu3;
  }
  say prime_count(1000000);
  say prime_count_approx(1000000);
  say nu3(1000000);

Construct and use a Sophie-Germain prime iterator:

  sub make_sophie_germain_iterator {
    my $p = shift || 2;
    my $it = prime_iterator($p);
    return sub {
      do { $p = $it->() } while !is_prime(2*$p+1);
      $p;
    };
  }
  my $sgit = make_sophie_germain_iterator();
  print $sgit->(), "\n"  for 1 .. 10000;

Project Euler, problem 3 (Largest prime factor):

  use Math::Prime::Util qw/factor/;
  use bigint;  # Only necessary for 32-bit machines.
  say 0+(factor(600851475143))[-1]

Project Euler, problem 7 (10001st prime):

  use Math::Prime::Util qw/nth_prime/;
  say nth_prime(10_001);

Project Euler, problem 10 (summation of primes):

  use Math::Prime::Util qw/sum_primes/;
  say sum_primes(2_000_000);
  #  ... or do it a little more manually ...
  use Math::Prime::Util qw/forprimes/;
  my $sum = 0;
  forprimes { $sum += $_ } 2_000_000;
  say $sum;
  #  ... or do it using a big list ...
  use Math::Prime::Util qw/vecsum primes/;
  say vecsum( @{primes(2_000_000)} );

Project Euler, problem 21 (Amicable numbers):

  use Math::Prime::Util qw/divisor_sum/;
  my $sum = 0;
  foreach my $x (1..10000) {
    my $y = divisor_sum($x)-$x;
    $sum += $x + $y if $y > $x && $x == divisor_sum($y)-$y;
  }
  say $sum;
  # Or using a pipeline:
  use Math::Prime::Util qw/vecsum divisor_sum/;
  say vecsum( map { divisor_sum($_) }
              grep { my $y = divisor_sum($_)-$_;
                     $y > $_ && $_==(divisor_sum($y)-$y) }
              1 .. 10000 );

Project Euler, problem 41 (Pandigital prime), brute force command line:

  perl -MMath::Prime::Util=primes,vecfirst -E 'say vecfirst { /1/&&/2/&&/3/&&/4/&&/5/&&/6/&&/7/} reverse @{primes(1000000,9999999)};'

Project Euler, problem 47 (Distinct primes factors):

  use Math::Prime::Util qw/pn_primorial factor_exp/;
  my $n = pn_primorial(4);  # Start with the first 4-factor number
  # factor_exp in scalar context returns the number of distinct prime factors
  $n++ while (factor_exp($n) != 4 || factor_exp($n+1) != 4 || factor_exp($n+2) != 4 || factor_exp($n+3) != 4);
  say $n;

Project Euler, problem 69, stupid brute force solution (about 1 second):

  use Math::Prime::Util qw/euler_phi/;
  my ($maxn, $maxratio) = (0,0);
  foreach my $n (1..1000000) {
    my $ndivphi = $n / euler_phi($n);
    ($maxn, $maxratio) = ($n, $ndivphi) if $ndivphi > $maxratio;
  }
  say "$maxn  $maxratio";

Here is the right way to do PE problem 69 (under 0.03s):

  use Math::Prime::Util qw/pn_primorial/;
  my $n = 0;
  $n++ while pn_primorial($n+1) < 1000000;
  say pn_primorial($n);

Project Euler, problem 187, stupid brute force solution, 1 to 2 minutes:

  use Math::Prime::Util qw/forcomposites factor/;
  my $nsemis = 0;
  forcomposites { $nsemis++ if scalar factor($_) == 2; } int(10**8)-1;
  say $nsemis;

Here is one of the best ways for PE187:  under 20 milliseconds from the
command line.  Much faster than Pari, and competitive with Mathematica.

  use Math::Prime::Util qw/forprimes prime_count/;
  my $limit = shift || int(10**8);
  $limit--;
  my ($sum, $pc) = (0, 1);
  forprimes {
    $sum += prime_count(int($limit/$_)) + 1 - $pc++;
  } int(sqrt($limit));
  say $sum;

To get the result of L<Math::Factor::XS/matches>:

  use Math::Prime::Util qw/divisors/;
  sub matches {
    my @d = divisors(shift);
    return map { [$d[$_],$d[$#d-$_]] } 1..(@d-1)>>1;
  }
  my $n = 139650;
  say "$n = ", join(" = ", map { "$_->[0]·$_->[1]" } matches($n));

or its C<matches> function with the C<skip_multiples> option:

  sub matches {
    my @d = divisors(shift);
    return map { [$d[$_],$d[$#d-$_]] }
           grep { my $div=$d[$_]; !scalar(grep {!($div % $d[$_])} 1..$_-1) }
           1..(@d-1)>>1; }
  }

Compute L<OEIS A054903|http://oeis.org/A054903> just like CRG4s Pari example:

  use Math::Prime::Util qw/forcomposites divisor_sum/;
  forcomposites {
    say if divisor_sum($_)+6 == divisor_sum($_+6)
  } 9,1e7;

Construct the table shown in L<OEIS A046147|http://oeis.org/A046147>:

  use Math::Prime::Util qw/znorder euler_phi gcd/;
  foreach my $n (1..100) {
    if (!znprimroot($n)) {
      say "$n -";
    } else {
      my $phi = euler_phi($n);
      my @r = grep { gcd($_,$n) == 1 && znorder($_,$n) == $phi } 1..$n-1;
      say "$n ", join(" ", @r);
    }
  }

Find the 7-digit palindromic primes in the first 20k digits of Pi:

  use Math::Prime::Util qw/Pi is_prime/;
  my $pi = "".Pi(20000);  # make sure we only stringify once
  for my $pos (2 .. length($pi)-7) {
    my $s = substr($pi, $pos, 7);
    say "$s at $pos" if $s eq reverse($s) && is_prime($s);
  }

  # Or we could use the regex engine to find the palindromes:
  while ($pi =~ /(([1379])(\d)(\d)\d\4\3\2)/g) {
    say "$1 at ",pos($pi)-7 if is_prime($1)
  }

The L<Bell numbers|https://en.wikipedia.org/wiki/Bell_number> B_n:

  sub B { my $n = shift; vecsum(map { stirling($n,$_,2) } 0..$n) }
  say "$_  ",B($_) for 1..50;

Recognizing tetrahedral numbers (L<OEIS A000292|http://oeis.org/A000292>):

  sub is_tetrahedral {
    my $n6 = vecprod(6,shift);
    my $k  = rootint($n6,3);
    vecprod($k,$k+1,$k+2) == $n6;
  }

Recognizing powerful numbers (e.g. C<ispowerful> from Pari/GP, or our
built-in and much faster L</is_powerful>):

  sub ispowerful { (vecall { $_->[1] > 1 } factor_exp(shift)) ? 1 : 0; }

Convert from binary to hex (3000x faster than Math::BaseConvert):

  my $hex_string = todigitstring(fromdigits($bin_string,2),16);

Calculate and print derangements using permutations:

  my @data = qw/a b c d/;
  forperm { say "@data[@_]" unless vecany { $_[$_]==$_ } 0..$#_ } @data;
  # Using forderange directly is faster

Compute the subfactorial of n (L<OEIS A000166|http://oeis.org/A000166>):

  sub my_subfactorial { my $n = shift;
    vecsum(map{ vecprod((-1)**($n-$_),binomial($n,$_),factorial($_)) }0..$n);
  }

Compute subfactorial (number of derangements) using simple recursion:

  sub my_subfactorial { my $n = shift;
    use bigint;
    ($n < 1)  ?  1  :  $n * subfactorial($n-1) + (-1)**$n;
  }

Recognize Sidon and sum-free sets.  We have specific functions
L</is_sidon_set> and L</is_sumfree_set> that are faster.

  sub is_sidon { my $set = shift;  my $len = scalar(@$set);
    my $sumset = sumset($set);
    0+(@$sumset==(($len*$len+$len)/2));
  }
  sub is_sum_free { my $set = shift;
    1 - setcontainsany($set,sumset($set));
  }



=head1 PRIMALITY TESTING NOTES

Above C<2^64>, L</is_prob_prime> performs an extra-strong
L<BPSW test|http://en.wikipedia.org/wiki/Baillie-PSW_primality_test>
which is fast (a little less than the time to perform 3 Miller-Rabin
tests) and has no known counterexamples.  If you trust the primality
testing done by Pari, Maple, SAGE, FLINT, etc., then this function
should be appropriate for you.  L</is_prime> will do the same BPSW
test as well as some additional testing, making it slightly more time
consuming but less likely to produce a false result.  This is a little
more stringent than Mathematica.  L</is_provable_prime> constructs a
primality proof.  If a certificate is requested, then either BLS75
theorem 5 or ECPP is performed.  Without a certificate, the method
is implementation specific (currently it is identical, but later
releases may use APRCL).  With L<Math::Prime::Util::GMP> installed,
this is quite fast through 300 or so digits.

Math systems 35 years ago typically used Miller-Rabin tests with C<k>
bases (usually fixed bases, sometimes random) for primality
testing, but these have generally been replaced by some form of BPSW
as used in this module.  See Pinch's 1993 paper for examples of why
using C<k> M-R tests leads to poor results.  All common contemporary
usage is now some BPSW variant.

=over 4

=item libtommath (previous to 1.1.0)

As of version 1.1.0 (January 2019), this uses strong BPSW and even adds
a base 3 strong pseudoprime test.  Raku uses this so fixes one of my
peeves I had with their design.

=item GMP/MPIR (previous to 6.2.0)

As of version 6.2.0 (January 2020), this uses strong BPSW and typically
adds one random-base strong pseudoprime test in addition.

=item L<Math::Pari> (previous to Pari 2.3.0)

Pari 2.1.7 is the default version installed with the L<Math::Pari>
module.  It uses 10 random M-R bases (the PRNG uses a fixed seed
set at compile time) and is highly susceptible to false positives.
Pari 2.3.0 was released in May 2006 and it uses BPSW (or the APR-CL
proof method), which are still used to this day in modern Pari/GP
(a great ECPP implementation was added in 2.10 for even better proofs).

=back

Basically the problem with running C<k> M-R tests is that it is too easy
to get counterexamples, forcing one to use a very large number of
tests (at least 20) to avoid frequent false results.  Using the BPSW test
results in no known counterexamples after 45+ years and runs much faster.
It can be enhanced with one or more random bases if one desires, and
will I<still> be much faster.


=head1 LIMITATIONS

Perl versions earlier than 5.8.0 have problems doing exact integer math.
Some operations will flip signs, and many operations will convert intermediate
or output results to doubles, which loses precision on 64-bit systems.
This causes numerous functions to not work properly.  The test suite will
try to determine if your Perl is broken (this only applies to really old
versions of Perl compiled for 64-bit when using numbers larger than
C<~ 2^49>).  The best solution is updating to a more recent Perl.

The module is thread-safe and should allow good concurrency on all platforms
that support Perl threads except Win32.  With Win32, either don't use threads
or make sure C<prime_precalc> is called before using C<primes>,
C<prime_count>, or C<nth_prime> with large inputs.  This is B<only>
an issue if you use non-Cygwin Win32 B<and> call these routines from within
Perl threads.

The block calls like L</forprimes>, L</vecreduce>, etc. use C<MULTICALL>.
We optimize away the per-call scope if it looks like it isn't needed.
This solves the functional and memory problems seen in
L<RT95409|https://rt.cpan.org/Ticket/Display.html?id=95409> and
L<RT127605|https://rt.cpan.org/Ticket/Display.html?id=127605>, while
still allowing higher performance on common simple blocks that don't
create temporary variables or pass local references out of scope.
Double braces for the function, e.g. C<forprimes {{ ... }} 50>, can be
used to force a separate scope.


=head1 SEE ALSO

This section describes other CPAN modules available that have some feature
overlap with this one.  Also see the L</REFERENCES> section.  Please let me
know if any of this information is inaccurate.  Also note that just because
a module doesn't match what I believe are the best set of features doesn't
mean it isn't perfect for someone else.

I will use SoE to indicate the Sieve of Eratosthenes, and MPU to denote this
module (L<Math::Prime::Util>).  Some quick alternatives I can recommend if
you don't want to use MPU:

=over 4

=item * L<Math::Prime::FastSieve> is the alternative module I use for basic
functionality with small integers.  It's fast and simple, and has a good
set of features.

=item * L<Math::Primality> is the alternative module I use for primality
testing on bigints.  The downside is that it can be slow, and the functions
other than primality tests are I<very> slow.

=item * L<Math::Pari> if you want the kitchen sink and can install it and
handle using it.  There are still some functions it doesn't do well
(e.g. prime count and nth_prime).

=back


L<Math::Prime::XS> has C<is_prime> and C<primes> functionality.  There is
no bigint support.  The C<is_prime> function uses well-written trial
division, meaning it is very fast for small numbers, but terribly slow for
large 64-bit numbers.  MPU is similarly fast with small numbers, but becomes
faster as the size increases.
MPXS's prime sieve is an unoptimized non-segmented SoE
which returns an array.  Sieve bases larger than C<10^7> start taking
inordinately long and using a lot of memory (gigabytes beyond C<10^10>).
E.g. C<primes(10**9, 10**9+1000)> takes 36 seconds with MPXS, but only
0.0001 seconds with MPU.

L<Math::Prime::FastSieve> supports C<primes>, C<is_prime>, C<next_prime>,
C<prev_prime>, C<prime_count>, and C<nth_prime>.  The caveat is that all
functions only work within the sieved range, so are limited to about C<10^10>.
It uses a fast SoE to generate the main sieve.  The sieve is 2-3x slower than
the base sieve for MPU, and is non-segmented so cannot be used for
larger values.  Since the functions work with the sieve, they are very fast.
The fast bit-vector-lookup functionality can be replicated in MPU using
C<prime_precalc> but is not required.

L<Bit::Vector> supports the C<primes> and C<prime_count> functionality in a
somewhat similar way to L<Math::Prime::FastSieve>.  It is the slowest of all
the XS sieves, and has the most memory use.  It is faster than pure Perl code.

L<Crypt::Primes> supports C<random_maurer_prime> functionality.  MPU has
more options for random primes (n-digit, n-bit, ranged, strong, and S-T) in
addition to Maurer's algorithm.  MPU does not have the critical bug
L<RT81858|https://rt.cpan.org/Ticket/Display.html?id=81858>.  MPU has
a more uniform distribution as well as returning a larger subset of primes
(L<RT81871|https://rt.cpan.org/Ticket/Display.html?id=81871>).
MPU does not depend on L<Math::Pari> though can run slow for bigints unless
the L<Math::BigInt::GMP> or L<Math::BigInt::Pari> modules are installed.
Having L<Math::Prime::Util::GMP> installed makes the speed vastly faster.
Crypt::Primes is hardcoded to use L<Crypt::Random> which uses /dev/random
(blocking source), while MPU uses its own ChaCha20 implementation seeded from
/dev/urandom or Win32.
MPU can return a primality certificate.
What Crypt::Primes has that MPU does not is the ability to return a generator.

L<Math::Factor::XS> calculates prime factors and factors, which correspond to
the L</factor> and L</divisors> functions of MPU.  Its functions do
not support bigints.  Both are implemented with trial division, meaning they
are very fast for really small values, but become very slow as the input
gets larger (factoring 19 digit semiprimes is over 1000 times slower).  The
function C<count_prime_factors> can be done in MPU using C<scalar factor($n)>.
See the L</"EXAMPLES"> section for a 2-line function replicating C<matches>.

L<Math::Big> version 1.16 includes features such as computing the
first C<n> primes, the first C<n> digits of Pi, base conversion, and
factorial.  With the latest versions it is reasonably fast for pure Perl.

L<Math::Big::Factors> supports factorization using wheel factorization (smart
trial division).  It supports bigints.  Unfortunately it is extremely slow on
any input that isn't the product of just small factors.  Even 7 digit inputs
can take hundreds or thousands of times longer to factor than MPU or
L<Math::Factor::XS>.  19-digit semiprimes will take I<hours> versus MPU's
single milliseconds.

L<Math::Factoring> is a placeholder module for bigint factoring.
The latest version, 0.02 from 2012, only correctly supports trial division.

L<Math::Prime::TiedArray> allows random access to a tied primes array, almost
identically to what MPU provides in L<Math::Prime::Util::PrimeArray>.  MPU
has attempted to fix Math::Prime::TiedArray's shift bug
(L<RT58151|https://rt.cpan.org/Ticket/Display.html?id=58151>).  MPU is
typically much faster and will use less memory, but there are some cases where
MP:TA is faster (MP:TA stores all entries up to the largest request, while
MPU:PA stores only a window around the last request).

L<List::Gen> is very interesting and includes a built-in primes iterator as
well as a C<is_prime> filter for arbitrary sequences.  Unfortunately both
are very slow.

L<Math::Primality> supports C<is_prime>, C<is_pseudoprime>,
C<is_strong_pseudoprime>, C<is_strong_lucas_pseudoprime>, C<next_prime>,
C<prev_prime>, C<prime_count>, and C<is_aks_prime> functionality.
This is a great little module that implements
primality functionality.  It was the first CPAN module to support the BPSW
test.  All inputs are processed using GMP, so it of course supports
bigints.  In fact, Math::Primality was made originally with bigints in mind,
while MPU was originally targeted to native integers, but both have added
better support for the other.  The main differences are extra functionality
(MPU has more functions) and performance.  With native integer inputs, MPU
is generally much faster, especially with L</prime_count>.  For bigints,
MPU is slower unless the L<Math::Prime::Util::GMP> module is installed, in
which case MPU is 2-4x faster.  L<Math::Primality> also installs
a C<primes.pl> program, but it has much less functionality than the one
included with MPU.

L<Math::NumSeq> does not have a one-to-one mapping between functions in MPU,
but it does offer a way to get many similar results such as
primes, twin primes, Sophie-Germain primes, lucky primes, moebius, divisor
count, factor count, Euler totient, primorials, etc.  Math::NumSeq is
set up for accessing these values in order rather than for arbitrary values,
though a few sequences support random access.  The primary advantage I see
is the uniform access mechanism for a I<lot> of sequences.  For those methods
that overlap, MPU is usually much faster.  Importantly, most of the sequences
in Math::NumSeq are limited to 32-bit indices.

L<Math::PlanePath::RationalsTree> enumerates fractions in various trees
including the Calkin-Wilf and Stern-Brocot trees.  All values must fit
in native integers.  There is a wealth of information in its documentation.

L<Math::ModInt::ChineseRemainder/cr_combine> is similar to MPU's L</chinese>,
and in fact they use the same algorithm.  The former module uses caching
of moduli to speed up further operations.  MPU does not do this.  This would
only be important for cases where the lcm is larger than a native int (noting
that use in cryptography would always have large moduli).

For combinations and permutations there are many alternatives.  One
difference with nearly all of them is that MPU's L</forcomb> and
L</forperm> functions don't operate directly on a user array but on
generic indices.
L<Math::Combinatorics> and L<Algorithm::Combinatorics> have more features,
but will be slower.
L<List::Permutor> does permutations with an iterator.
L<Algorithm::FastPermute> and L<Algorithm::Permute> are very similar
but can be 2-10x faster than MPU (they use the same user-block
structure but twiddle the user array each call).

There are numerous modules to perform a set product (also called Cartesian
product or cross product).  These include L<Set::Product>,
L<Math::Cartesian::Product>, L<Set::Scalar>, and L<Set::CrossProduct>,
as well as a few others.
The L<Set::CartesianProduct::Lazy> module provides random access,
albeit rather slowly.
Our L</forsetproduct> matches L<Set::Product::XS> in both high performance
and functionality (that module was written earlier, and our function is
nearly identical to L<Set::Product::XS/product>).

L<Math::Pari> supports a lot of features, with a great deal of overlap.  In
general, MPU will be faster for native 64-bit integers, while it differs
for bigints (Pari will always be faster if L<Math::Prime::Util::GMP> is not
installed; with it, it varies by function).  Note that Pari extends many of
these functions to other spaces (Gaussian integers, complex numbers, vectors,
matrices, polynomials, etc.) which are beyond the realm of this module.
Some of the highlights:

=over 4

=item C<isprime>

The default L<Math::Pari> is built with Pari 2.1.7.  This uses 10 M-R
tests with randomly chosen bases (fixed seed, but doesn't reset each
invocation like GMP's C<is_probab_prime>).  This has a much greater chance
of false positives compared to the BPSW test -- some composites such as
C<9>, C<88831>, C<38503>, etc.
(L<OEIS A141768|http://oeis.org/A141768>)
have a surprisingly high chance of being indicated prime.
Using C<isprime($n,1)> will perform an C<n-1> proof,
but this becomes unreasonably slow past 70 or so digits.

If L<Math::Pari> is built using Pari 2.3.5 (this requires manual
configuration) then the primality tests are completely different.  Using
C<ispseudoprime> will perform a BPSW test and is quite a bit faster than
the older test.  C<isprime> now does an APR-CL proof (fast, but no
certificate).

L<Math::Primality> uses a strong BPSW test, which is the standard BPSW
test based on the 1980 paper.  It has no known counterexamples (though
like all these tests, we know some exist).  Pari 2.3.5 (and through at
least 2.6.2) uses an almost-extra-strong BPSW test for its
C<ispseudoprime> function.  This is deterministic for native integers,
and should be excellent for bigints, with a slightly lower chance of
counterexamples than the traditional strong test.
L<Math::Prime::Util> uses the
full extra-strong BPSW test, which has an even lower chance of
counterexample.
With L<Math::Prime::Util::GMP>, C<is_prime> adds an extra M-R test
using a random base, which further reduces the probability of a composite
being allowed to pass.

=item C<primepi>

Only available with version 2.3 of Pari.  Similar to MPU's L</prime_count>
function in API, but uses a naive counting algorithm with its precalculated
primes, so is not of practical use.  Incidentally, Pari 2.6 (not usable from
Perl) has fixed the pre-calculation requirement so it is more useful, but is
still thousands of times slower than MPU.

=item C<primes>

Doesn't support ranges, requires bumping up the precalculated
primes for larger numbers, which means knowing in advance the upper limit
for primes.  Support for numbers larger than 400M requires using Pari
version 2.3.5.  If that is used, sieving is about 2x faster than MPU, but
doesn't support segmenting.

=item C<factorint>

Similar to MPU's L</factor_exp> though with a slightly different return.
MPU offers L</factor> for a linear array of prime factors where
   n = p1 * p2 * p3 * ...   as (p1,p2,p3,...)
and L</factor_exp> for an array of factor/exponent pairs where:
   n = p1^e1 * p2^e2 * ...  as ([p1,e1],[p2,e2],...)
Pari/GP returns an array similar to the latter.  L<Math::Pari> returns
a transposed matrix like:
   n = p1^e1 * p2^e2 * ...  as ([p1,p2,...],[e1,e2,...])
Slower than MPU for all 64-bit inputs on an x86_64 platform, it may be
faster for large values on other platforms.  With the newer
L<Math::Prime::Util::GMP> releases, bigint factoring is slightly
faster on average in MPU.

=item C<divisors>

Similar to MPU's L</divisors>.

=item C<forprime>, C<forcomposite>, C<fordiv>, C<sumdiv>

Similar to MPU's L</forprimes>, L</forcomposites>, L</fordivisors>, and
L</divisor_sum>.

=item C<eulerphi>, C<moebius>

Similar to MPU's L</euler_phi> and L</moebius>.  MPU is 2-20x faster for
native integers.  MPU also supported range inputs, which can be much
more efficient.  With bigint arguments, MPU is slightly faster than
Math::Pari if the GMP backend is available, but very slow without.

=item C<gcd>, C<lcm>, C<kronecker>, C<znorder>, C<znprimroot>, C<znlog>

Similar to MPU's L</gcd>, L</lcm>, L</kronecker>, L</znorder>,
L</znprimroot>, and L</znlog>.  Pari's C<znprimroot> only returns the
smallest root for prime powers.  The behavior is undefined when the group is
not cyclic (sometimes it throws an exception, sometimes it returns
an incorrect answer, sometimes it hangs).  MPU's L</znprimroot> will always
return the smallest root if it exists, and C<undef> otherwise.
Similarly, MPU's L</znlog> will return the smallest C<x> and work with
non-primitive-root C<g>, which is similar to Pari/GP 2.6, but not the
older versions in L<Math::Pari>.  The performance of L</znlog> is quite
good compared to older Pari/GP, but much worse than 2.6's new methods.

=item C<sigma>

Similar to MPU's L</divisor_sum>.  MPU is ~10x faster when the result
fits in a native integer.  Once things overflow it is fairly similar in
performance.  However, using L<Math::BigInt> can slow things down quite
a bit, so for best performance in these cases using a L<Math::GMP> object
is best.

=item C<numbpart>, C<forpart>

Similar to MPU's L</partitions> and L</forpart>.  These functions were
introduced in Pari 2.3 and 2.6, hence are not in Math::Pari.  C<numbpart>
produces identical results to C<partitions>, but Pari is I<much> faster.
L</forpart> is very similar to Pari's function, but produces a different
ordering (MPU is the standard anti-lexicographical, Pari uses a size sort).
Currently Pari is somewhat faster due to Perl function call overhead.  When
using restrictions, Pari has much better optimizations.

=item C<eint1>

Similar to MPU's L</ExponentialIntegral>.

=item C<zeta>

MPU has L</RiemannZeta> which takes non-negative real inputs, while Pari's
function supports negative and complex inputs.

=back

Overall, L<Math::Pari> supports a huge variety of functionality and has a
sophisticated and mature code base behind it (noting that the Pari library
used is about 10 years old now).
For native integers, typically Math::Pari will be slower than MPU.  For
bigints, Math::Pari may be superior and it rarely has any performance
surprises.  Some of the
unique features MPU offers include super fast prime counts, nth_prime,
ECPP primality proofs with certificates, approximations and limits for both,
random primes, fast Mertens calculations, Chebyshev theta and psi functions,
and the logarithmic integral and Riemann R functions.  All with fairly
minimal installation requirements.

For Python, the package L<labmath|https://pypi.org/project/labmath/> looks
to have similar overall goals.


=head1 PERFORMANCE

First, for those looking for the state of the art non-Perl solutions:

=over 4

=item Primality testing

For general numbers smaller than 2000 or so digits, MPU is the fastest
solution I am aware of (it is faster than Pari 2.7, PFGW, and FLINT).
For very large inputs,
L<PFGW|http://sourceforge.net/projects/openpfgw/> is the fastest primality
testing software I'm aware of.  It has fast trial division, and is especially
fast on many special forms.  It does not have a BPSW test however, and there
are quite a few counterexamples for a given base of its PRP test, so it is
commonly used for fast filtering of large candidates.
A test such as the BPSW test in this module is then recommended.

=item Primality proofs

L<Primo|http://www.ellipsa.eu/> is the best method for open source primality
proving for inputs over 1000 digits.  Primo also does well below that size,
but other good alternatives are
David Cleaver's L<mpzaprcl|http://sourceforge.net/projects/mpzaprcl/>,
the APRCL from the modern L<Pari|http://pari.math.u-bordeaux.fr/> package,
or the standalone ECPP from this module with large polynomial set.

=item Factoring

L<yafu|http://sourceforge.net/projects/yafu/>,
L<msieve|http://sourceforge.net/projects/msieve/>, and
L<gmp-ecm|http://ecm.gforge.inria.fr/> are all good choices for large
inputs.  The factoring code in this module (and all other CPAN modules) is
very limited compared to those.

=item Primes

L<primesieve|http://code.google.com/p/primesieve/> and
L<yafu|http://sourceforge.net/projects/yafu/>
are the fastest publicly available code I am aware of.  Primesieve
will additionally take advantage of multiple cores with excellent
efficiency.
Tomás Oliveira e Silva's private code may be faster for very large
values, but isn't available for testing.

Note that the Sieve of Atkin is I<not> faster than the Sieve of Eratosthenes
when both are well implemented.  The only Sieve of Atkin that is even
competitive is Bernstein's super optimized I<primegen>, which runs on par
with the SoE in this module.  The SoE's in Pari, yafu, and primesieve
are all faster.

=item Prime Counts and Nth Prime

The gold standard is currently Kim Walisch's fantastic
L<primecount|https://github.com/kimwalisch/primecount>.
For single threaded computations with 64-bit C<n>, this module is fairly
close in performance.

The fastest solution for small inputs is a hybrid table/sieve method.
This module does this for values below 60M.  As the inputs get larger,
either the tables have to grow exponentially or speed must be
sacrificed, so eventually we will use methods like LMO.

=back


=head2 PRIME COUNTS

Counting the primes to C<800_000_000> (800 million):

  Time (s)   Module                      Version  Notes
  ---------  --------------------------  -------  -----------
       0.001 Math::Prime::Util           0.37     using extended LMO
       0.007 Math::Prime::Util           0.12     using Lehmer's method
       0.27  Math::Prime::Util           0.17     segmented mod-30 sieve
       0.39  Math::Prime::Util::PP       0.24     Perl (Lehmer's method)
       2.9   Math::Prime::FastSieve      0.12     decent odd-number sieve
      11.7   Math::Prime::XS             0.27     0.27 includes a count
      15.0   Bit::Vector                 7.2
      48.9   Math::Prime::Util::PP       0.14     Perl (fastest I know of)
      49.00  Math::Big                   1.16     Uses efficient Perl
     170.0   Faster Perl sieve (net)     2012-01  array of odds
     548.1   RosettaCode sieve (net)     2012-06  simplistic Perl
    3048.1   Math::Primality             0.08     Perl + Math::GMPz

Python's SymPy 1.1 (2017) up to current 1.14.0 (2025) uses Legendre's method.
This is vastly preferable to sieving used by earlier versions of SymPy and by
MPMATH (as of v1.4.0).  It is a little slower than our Lehmer and quite a bit
slower than LMO, but is much simpler.

=head2 PRIMALITY TESTING

=over 4

=item Small inputs:  is_prime from 1 to 20M

    2.0s  Math::Prime::Util      (sieve lookup if prime_precalc used)
    2.5s  Math::Prime::FastSieve (sieve lookup)
    3.3s  Math::Prime::Util      (trial + deterministic M-R)
   10.4s  Math::Prime::XS        (trial)
   19.1s  Math::Pari w/2.3.5     (BPSW)
   52.4s  Math::Pari             (10 random M-R)
  480s    Math::Primality        (deterministic M-R)

=item Large native inputs:  is_prime from 10^16 to 10^16 + 20M

    4.5s  Math::Prime::Util      (BPSW)
   24.9s  Math::Pari w/2.3.5     (BPSW)
  117.0s  Math::Pari             (10 random M-R)
  682s    Math::Primality        (BPSW)
  30 HRS  Math::Prime::XS        (trial)

  These inputs are too large for Math::Prime::FastSieve.

=item bigints:  is_prime from 10^100 to 10^100 + 0.2M

    2.2s  Math::Prime::Util          (BPSW + 1 random M-R)
    2.7s  Math::Pari w/2.3.5         (BPSW)
   13.0s  Math::Primality            (BPSW)
   35.2s  Math::Pari                 (10 random M-R)
   38.6s  Math::Prime::Util w/o GMP  (BPSW)
   70.7s  Math::Prime::Util          (n-1 or ECPP proof)
  102.9s  Math::Pari w/2.3.5         (APR-CL proof)

=back

=over 4

=item *

MPU is consistently the fastest solution, and performs the most
stringent probable prime tests on bigints.

=item *

Math::Primality has a lot of overhead that makes it quite slow for
native size integers.  With bigints we finally see it work well.

=item *

Math::Pari built with 2.3.5 not only has a better primality test versus
the default 2.1.7, but runs faster.  It still has quite a bit of overhead
with native size integers.  Pari/GP 2.5.0 takes 11.3s, 16.9s, and 2.9s
respectively for the tests above.  MPU is still faster, but clearly the
time for native integers is dominated by the calling overhead.

=back

=head2 FACTORING

Factoring performance depends on the input, and the algorithm choices used
are still being tuned.  L<Math::Factor::XS> is very fast when given input with
only small factors, but it slows down rapidly as the smallest factor increases
in size.  For numbers larger than 32 bits, L<Math::Prime::Util> can be 100x or
more faster (a number with only very small factors will be nearly identical,
while a semiprime may be 3000x faster).  L<Math::Pari>
is much slower with native sized inputs, probably due to calling
overhead.  For bigints, the L<Math::Prime::Util::GMP> module is needed or
performance will be far worse than Math::Pari.  With the GMP module,
performance is pretty similar from 20 through 70 digits, with the caveat
that the current MPU factoring uses more memory for 60+ digit numbers.


L<This slide presentation|http://math.boisestate.edu/~liljanab/BOISECRYPTFall09/Jacobsen.pdf>
has a lot of data on 64-bit and GMP factoring performance I collected in 2009.
Assuming you do not know anything about the inputs, trial division and
optimized Fermat or Lehman work very well for small numbers (<= 10 digits),
while native SQUFOF is typically the method of choice for 11-18 digits (I've
seen claims that a lightweight QS can be faster for 15+ digits).  Some form
of Quadratic Sieve is usually used for inputs in the 19-100 digit range, and
beyond that is the General Number Field Sieve.  For serious factoring,
I recommend looking at
L<yafu|http://sourceforge.net/projects/yafu/>,
L<msieve|http://sourceforge.net/projects/msieve/>,
L<gmp-ecm|http://ecm.gforge.inria.fr/>,
L<GGNFS|http://sourceforge.net/projects/ggnfs/>,
and L<Pari|http://pari.math.u-bordeaux.fr/>.  The latest yafu should cover most
uses, with GGNFS likely only providing a benefit for numbers large enough to
warrant distributed processing.

=head2 PRIMALITY PROVING

The C<n-1> proving algorithm in L<Math::Prime::Util::GMP> compares well to
the version included in Pari.  Both are pretty fast to about 60 digits, and
work reasonably well to 80 or so before starting to take many minutes per
number on a fast computer.  Version 0.09 and newer of MPU::GMP contain an
ECPP implementation that, while not state of the art compared to closed source
solutions, works quite well.
It averages less than a second for proving 200-digit primes
including creating a certificate.  Times below 200 digits are faster than
Pari 2.3.5's APR-CL proof.  For larger inputs the bottleneck is a limited set
of discriminants, and time becomes more variable.  There is a larger set of
discriminants on github that help, with 300-digit primes taking ~5 seconds on
average and typically under a minute for 500-digits.  For primality proving
with very large numbers, I recommend L<Primo|http://www.ellipsa.eu/>.

=head2 RANDOM PRIME GENERATION

Seconds per prime for random prime generation on an early 2015 Macbook Pro
(2.7 GHz i5) with L<Math::BigInt::GMP> and L<Math::Prime::Util::GMP> installed.

  bits    random   +testing   Maurer   Shw-Tylr  CPMaurer
  -----  --------  --------  --------  --------  --------
     64    0.00002 +0.000009   0.00004   0.00004    0.019
    128    0.00008 +0.00014    0.00018   0.00012    0.051
    256    0.0004  +0.0003     0.00085   0.00058    0.13
    512    0.0023  +0.0007     0.0048    0.0030     0.40
   1024    0.019   +0.0033     0.034     0.025      1.78
   2048    0.26    +0.014      0.41      0.25       8.02
   4096    2.82    +0.11       4.4       3.0      66.7
   8192   23.7     +0.65      50.8      38.7     929.4

  random    = random_nbit_prime  (results pass BPSW)
  random+   = additional time for 3 M-R and a Frobenius test
  maurer    = random_maurer_prime
  Shw-Tylr  = random_shawe_taylor_prime
  CPMaurer  = Crypt::Primes::maurer

L</random_nbit_prime> is reasonably fast, and for most purposes should
suffice.  For cryptographic purposes, one may want additional tests or
a proven prime.  Additional tests are quite cheap, as shown by the time
for three extra M-R and a Frobenius test.  At these bit sizes, the
chances a composite number passes BPSW, three more M-R tests, and a
Frobenius test is I<extraordinarily> small.

L</random_proven_prime> provides a randomly selected prime with an optional
certificate, without specifying the particular method.  With GMP installed
this always uses Maurer's algorithm as it is the best compromise between
speed and diversity.

L</random_maurer_prime> constructs a provable prime.  A primality test is
run on each intermediate, and it also constructs a complete primality
certificate which is verified at the end (and can be returned).  While the
result is uniformly distributed, only about 10% of the primes in the range
are selected for output.  This is a result of the FastPrime algorithm and
is usually unimportant.

L</random_shawe_taylor_prime> similarly constructs a provable prime.  It
uses a simpler construction method.  It is slightly faster than Maurer's
algorithm but provides less diversity (even fewer primes in the range are
selected, though for typical cryptographic sizes this is not important).
The Perl implementation uses a single large random seed followed by
SHA-256 as specified by FIPS 186-4.  The GMP implementation uses the same
FIPS 186-4 algorithm but uses its own CSPRNG which may not be SHA-256.

L<Crypt::Primes/maurer> times are included for comparison.  It is reasonably
fast for small sizes but gets slow as the size increases.  It is 10 to 500
times slower than this module's GMP methods.  It does not
perform any primality checks on the intermediate results or the final
result (I highly recommended running a primality test on the output).
Additionally important for servers, L<Crypt::Primes/maurer> uses excessive
system entropy and can grind to a halt if C</dev/random> is exhausted
(it can take B<days> to return).

=head2 CONGRUENT NUMBERS

The L</is_congruent_number> function, combined with our L</forsquarefreeint>
operator to loop over square free integers in a range, is quite fast
compared to most public implementations.
For computing many values, it is expected that fast theta series
computations, such as demonstrated in
Hart et al. (2009) (L<https://wrap.warwick.ac.uk/id/eprint/41654/>),
are significantly faster, albeit requiring more memory and disk space.

All congruent numbers less than 300,000 can be identified in under 2 seconds.

Giovanni Resta's list of 213318 square-free and C<mod 8 <= 4>
congruent numbers less than C<10^7> can be generated in 19 minutes on
a single core of an M1 laptop.



=head2 SETS

Measuring the performance of various modules for set operations doesn't
give a strict order.  Many modules are fast at some operations and slow
at others.  Some have particular inputs they are very fast or very slow
with.  Each module has different functionality.

We chose, following Pari and Mathematica, to represent sets as native
Perl lists of sorted de-duplicated integers, rather than a dedicated object.
This allows flexibility and use for other purposes, but it isn't ideal
for general performance, especially with very large sets (100k+ elements)
where we spend a large amount of time parsing and manipulating the
Perl input array.  While an opaque data structure would use 8 or fewer
bytes per element, Perl arrays use approximately 32 bytes per integer.
Still, this is quite favorable compared to Perl hashes at 120 to 220
(e.g. Set::Light, Set::Tiny, Set::Scalar, Set::Functional).

For generic set use, I recommend L<Set::Tiny>.
The module source is B<very> tiny, unlike this module.
It offers an easy API for basic set functions and is fast.
It is not limited to integers.
On the other hand, with integers our module is typically faster (2-10x)
and uses less memory, even with our choice of native Perl sorted arrays.

Finding the sumset size of the first 10,000 primes.

  my %r;  my $p = primes(nth_prime(10000));

  12.6s   15MB  forsetproduct {$r{vecsum(@_)}=undef;} $p,$p;
                say scalar(keys %r);
   9.4s 3900MB  Pari/GP X=primes(10000); #setbinop((a,b)->a+b,X,X)
   2.4s    3MB  $s=setbinop { $a+$b } $p;  say scalar @$s;
   0.4s    3MB  $s=sumset $p;  say scalar @$s;

Set intersection of C<[-1000..100]> and C<[-100..1000]>, with Perl 5.43.7.

     4 uS  Set::IntSpan::Fast::XS
     5 uS  setintersect                       <===========  this module
     7 uS  Pari/GP 2.17.0
    14 uS  Set::IntSpan::Fast
    61 uS  native Perl hash intersection          /\ /\ /\  Faster
    62 uS  Set::Tiny
    66 uS  Set::Functional
   105 uS  PP::setintersect                       \/ \/ \/  Slower
   200 uS  Array::Set
   310 uS  Set::Object
   332 uS  Set::SortedArray
  1508 uS  Set::Scalar

Set intersection of integers under 1000 divisible by 2 and 3 respectively.
Sets are C<[grep{0==$_%2}0..999]> and C<[grep{0==$_%3}0..999]>:

     3 uS  setintersect                       <===========  this module
     6 uS  Pari/GP 2.17.0
    31 uS  Set::Tiny
    32 uS  native Perl hash intersection          /\ /\ /\  Faster
    34 uS  Set::Functional
    37 uS  PP::setintersect                       \/ \/ \/  Slower
    64 uS  Set::IntSpan::Fast::XS
    86 uS  Array::Set
   122 uS  Set::SortedArray
   138 uS  Set::Object
   615 uS  Set::Scalar
  3090 uS  Set::IntSpan::Fast

Set::IntSpan::Fast is very fast with the first example using single span
sets, but gets quite slow with more spans as seen in the second example.
The other modules are mostly unaffected by data patterns.

Using our own set objects wrapping a C structure of some sort would be
faster and lower memory.  In particular, we often spend more time just
reading the set values than we do performing the set operation.

=head2 SORTING

Perl's built-in sort is a cache-friendly stable merge sort.  This is
reasonably appropriate for the wide variety of uses expected.  When sorting
lists of integers, it could be improved.  Perl 5.8 brought an in-place
optimization, so C<@a=sort{$a<=>$b}@a> is done without copying.  The
numerical sort is recognized and short-cut so doesn't actually call the
well-known comparison function.  However, Perl's old 32-bit legacy lived
on until 5.26 as the inputs were turned into doubles, which can lead to
subtle bugs with large integers.  Inputs that started as strings (e.g. input
read from a file) will still get turned into doubles.

Our vecsort tries to avoid these issues, making sure inputs are processed as
only IV, UV, and/or bigints.  Integer strings are converted to one of those.
All inputs are validated to be integers.  There is no need for separate
interfaces for signed and unsigned numbers as Perl's representation stores
this information explicitly and per-variable rather than per-array.

Input lists that contain bigints, or both negative numbers and positive
numbers larger than the maximum IV (C<2^63-1> for 64-bit), cannot be stored
in a native array of a single type, therefore will be sorted using Perl's
sort rather than our C code.  This is B<substantially> slower, but produces
the correct results.

Our sorting for native signed and unsigned integers is a combination of
radix sort and quicksort (the latter using median of 9 partitioning,
insertion sort for small partitions, and heapsort fallback if we detect
repeated poor partitioning).  It is quite fast and low overhead.

L<Sort::XS> has a variety of algorithms.
However there is no option for unsigned (UV), only signed integers (IV).
Sort::Key offers a variety of interfaces including unsigned and signed
integers, as well as in-place versions.
The following table compares sorting random 64-bit unsigned integers and
is shown as speedup relative to Perl's sort (higher is faster, v5.43.7).

                              10    100   1000  10000 100000     1M
  vecsort                   2.0x   2.3x   4.7x   6.2x   6.9x   9.7x
  Sort::Key::Radix usort    1.4x   2.3x   3.4x   3.4x   4.7x   2.7x
  Sort::XS::quick_sort      1.2x   1.5x   1.8x   1.9x   2.0x   2.7x
  Sort::Key usort           1.2x   1.3x   1.3x   1.3x   1.3x   1.3x
  sort                      1.0x   1.0x   1.0x   1.0x   1.0x   1.0x
  List::MoreUtils::qsort    0.6x   0.5x   0.4x   0.4x   0.4x   0.3x

The implementation does not currently try to exploit patterns.
Regarding the above timing, when given sorted or reverse sorted data,
Perl's sort is much faster versus the random values used above, though
still not faster than L</vecsort> and L<Sort::Key::Radix::usort> (both
of which use a radix sort).

List::MoreUtils::qsort has very different goals in mind than standard
sorting of integer lists, as mentioned in their documentation.
In contrast, this is exactly (and only) what vecsort does, so it should
not be a surprise that our function looks good on this benchmark.
Different use cases would show things differently.


=head1 AUTHORS

Dana Jacobsen E<lt>dana@acm.orgE<gt>


=head1 ACKNOWLEDGEMENTS

Eratosthenes of Cyrene provided the elegant and simple algorithm for finding
primes.

Terje Mathisen, A.R. Quesada, B. Van Pelt, and Kim Walisch all had useful
ideas I used in my wheel sieve.

The SQUFOF implementation being used is a slight modification to the public
domain racing version written by Ben Buhrow.  Enhancements with ideas from
Ben's later code as well as Jason Papadopoulos's public domain implementations
are planned for a later version.

The LMO implementation is based on the 2003 preprint from Christian Bau,
as well as the 2006 paper from Tomás Oliveira e Silva.  I also want to
thank Kim Walisch for the many discussions about prime counting.


=head1 REFERENCES

=over 4

=item *

Christian Axler, "New bounds for the prime counting function π(x)", September 2014.  For large values, improved limits versus Dusart 2010.  L<http://arxiv.org/abs/1409.1780>

=item *

Christian Axler, "Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen", January 2013.  Prime count and nth-prime bounds in more detail.  Thesis in German, but first part is easily read.  L<http://docserv.uni-duesseldorf.de/servlets/DerivateServlet/Derivate-28284/pdfa-1b.pdf>

=item *

Christian Bau, "The Extended Meissel-Lehmer Algorithm", 2003, preprint with example C++ implementation.  Very detailed implementation-specific paper which was used for the implementation here.  Highly recommended for implementing a sieve-based LMO.  L<http://cs.swan.ac.uk/~csoliver/ok-sat-library/OKplatform/ExternalSources/sources/NumberTheory/ChristianBau/>

=item *

Manuel Benito and Juan L. Varona, "Recursive formulas related to the summation of the Möbius function", I<The Open Mathematics Journal>, v1, pp 25-34, 2007.  Among many other things, shows a simple formula for computing the Mertens functions with only n/3 Möbius values (not as fast as Deléglise and Rivat, but really simple).  L<http://www.unirioja.es/cu/jvarona/downloads/Benito-Varona-TOMATJ-Mertens.pdf>

=item *

John Brillhart, D. H. Lehmer, and J. L. Selfridge, "New Primality Criteria and Factorizations of 2^m +/- 1", Mathematics of Computation, v29, n130, Apr 1975, pp 620-647.  L<http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf>

=item *

W. J. Cody and Henry C. Thacher, Jr., "Rational Chebyshev Approximations for the Exponential Integral E_1(x)", I<Mathematics of Computation>, v22, pp 641-649, 1968.

=item *

W. J. Cody and Henry C. Thacher, Jr., "Chebyshev approximations for the exponential integral Ei(x)", I<Mathematics of Computation>, v23, pp 289-303, 1969.  L<http://www.ams.org/journals/mcom/1969-23-106/S0025-5718-1969-0242349-2/>

=item *

W. J. Cody, K. E. Hillstrom, and Henry C. Thacher Jr., "Chebyshev Approximations for the Riemann Zeta Function", I<Mathematics of Computation>, v25, n115, pp 537-547, July 1971.

=item *

Henri Cohen, "A Course in Computational Algebraic Number Theory", Springer, 1996.  Practical computational number theory from the team lead of L<Pari|http://pari.math.u-bordeaux.fr/>.  Lots of explicit algorithms.

=item *

Marc Deléglise and Joöl Rivat, "Computing the summation of the Möbius function", I<Experimental Mathematics>, v5, n4, pp 291-295, 1996.  Enhances the Möbius computation in Lioen/van de Lune, and gives a very efficient way to compute the Mertens function.  L<http://projecteuclid.org/euclid.em/1047565447>

=item *

Pierre Dusart, "Autour de la fonction qui compte le nombre de nombres premiers", PhD thesis, 1998.  In French.  The mathematics is readable and highly recommended reading if you're interested in prime number bounds.  L<http://www.unilim.fr/laco/theses/1998/T1998_01.html>

=item *

Pierre Dusart, "Estimates of Some Functions Over Primes without R.H.", preprint, 2010.  Updates to the best non-RH bounds for prime count and nth prime.  L<http://arxiv.org/abs/1002.0442/>

=item *

Pierre-Alain Fouque and Mehdi Tibouchi, "Close to Uniform Prime Number Generation With Fewer Random Bits", pre-print, 2011.  Describes random prime distributions, their algorithm for creating random primes using few random bits, and comparisons to other methods.  Definitely worth reading for the discussions of uniformity.  L<http://eprint.iacr.org/2011/481>

=item *

Daan Leijen, "Division and Modulus for Computer Scientists", 2001.  Paper discussing different div/mod methods.  L<https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/divmodnote-letter.pdf>

=item *

Walter M. Lioen and Jan van de Lune, "Systematic Computations on Mertens' Conjecture and Dirichlet's Divisor Problem by Vectorized Sieving", in I<From Universal Morphisms to Megabytes>, Centrum voor Wiskunde en Informatica, pp. 421-432, 1994.  Describes a nice way to compute a range of Möbius values.  L<http://walter.lioen.com/papers/LL94.pdf>

=item *

Ueli M. Maurer, "Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters", 1995.  Generating random provable primes by building up the prime.  L<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.2151>

=item *

Gabriel Mincu, "An Asymptotic Expansion", I<Journal of Inequalities in Pure and Applied Mathematics>, v4, n2, 2003.  A very readable account of Cipolla's 1902 nth prime approximation.  L<http://www.emis.de/journals/JIPAM/images/153_02_JIPAM/153_02.pdf>

=item *

L<OEIS: Primorial|http://oeis.org/wiki/Primorial>

=item *

Vincent Pegoraro and Philipp Slusallek, "On the Evaluation of the Complex-Valued Exponential Integral", I<Journal of Graphics, GPU, and Game Tools>, v15, n3, pp 183-198, 2011.  L<http://www.cs.utah.edu/~vpegorar/research/2011_JGT/paper.pdf>

=item *

William H. Press et al., "Numerical Recipes", 3rd edition.

=item *

Hans Riesel, "Prime Numbers and Computer Methods for Factorization", Birkh?user, 2nd edition, 1994.  Lots of information, some code, easy to follow.

=item *

David M. Smith, "Multiple-Precision Exponential Integral and Related Functions", I<ACM Transactions on Mathematical Software>, v37, n4, 2011.  L<http://myweb.lmu.edu/dmsmith/toms2011.pdf>

=item *

Douglas A. Stoll and Patrick Demichel , "The impact of ζ(s) complex zeros on π(x) for x E<lt> 10^{10^{13}}", I<Mathematics of Computation>, v80, n276, pp 2381-2394, October 2011.  L<http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02477-4/home.html>

=back


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