Math::Big::Factors - factor big numbers into prime factors using different algorithmns

    use Math::Big::Factors;

    $wheel	= wheel (4);			# prime number wheel of 2,3,5,7
    print $wheel->[0],$wheel->[1],$wheel->[2],$wheel->[3],"\n";

    @factors	= factor_wheel(19*71*59*3,1);	# using prime wheel of order 1
    @factors	= factor_wheel(19*71*59*3,7);	# using prime wheel of order 7

	$wheel = wheel($o);

#############################################################################
# Math/Big/Factors.pm -- factor big numbers into prime factors

package Math::Big::Factors;
use vars qw($VERSION);
$VERSION = 1.02;    # Current version of this package
require  5.006002;  # requires this Perl version or later

use Math::BigInt;
use Math::BigFloat;
use Math::Big;
use Exporter;
@ISA = qw( Exporter );
@EXPORT_OK = qw( wheel factors_wheel
               );
#@EXPORT = qw( );
use strict;

sub wheel
  {
  # calculate a prime-wheel of order $o
  my $o = abs(shift || 1);              # >= 1

  # some primitive wheels as shortcut:
  return [ Math::BigInt->new(2), Math::BigInt->new(1) ] if $o == 1;

  my @primes = Math::Big::primes($o*5);		# initial primes, get some more

  my $mul = Math::BigInt->new(1); my @wheel;
  for (my $i = 0; $i < $o; $i++)
    {
    #print "$primes[$i]\n";
    $mul *= $primes[$i]; push @wheel,$primes[$i];
    }
  #print "Mul $mul\n";
  my $last = $wheel[-1];                        # get biggest initial
  #print "last is $last\n";
  # now sieve any number that is a multiply of one of the inital ones
  @primes = ();                                 # undef => leftover
  foreach (@wheel)
    {
    next if $_ == 2;                            # dont mark these, we skip 'em
    my $i = $_; my $add = $i;
    while ($i < $mul)
      {
      $primes[$i] = 1; $i += $add;
      }
    }
  push @wheel, Math::BigInt->new(1);
  my $i = $last;
  while ($i < $mul)
    {
    push @wheel,$i if !defined $primes[$i]; $i += 2;    # skip even ones
    }
  \@wheel;
  }
 
sub _transform_wheel
  {
  # from a given prime-wheel, calculate a increment table that can be used
  # to step trough numbers
  # input:  ref to array with prime wheel
  # output: ($restart,$ref_to_add_table);

  my (@wheel,$we);
  my $add = shift; shift @$add;         # remove the first 2 from wheel

  if (@$add == 1)                       # order 1
    {
    my $two = Math::BigInt->new(2);
    # (2,2) or (2,2,2,2,2,2) etc would do, too
    @wheel = ($two->copy(),$two->copy(),$two->copy(),$two->copy());
    return (1,\@wheel);
    }
  # from the list of divisors above create a add-table which we can take to
  # increment from 3 onwards. The tabe consists of two parts, the second part
  # will be repeatedly used
  my $last = -1; my $mod = 2; my $i = 0;
  # create the increment part for the initial primes (3,5, or 3,5,7 etc)
  while ($add->[$i] != 1)
    {
    $mod *= $add->[$i];
    push @wheel, $add->[$i] - $last if $last != -1;     # skip the first
    #print $wheel[-1],"\n" if $last != -1;
    $last = $add->[$i]; $i++;
    }
  #print "mod $mod\n";
  my $border = $i-1;                                    # account for ++
  my $length = scalar @$add-$i;
  my $ws = $border+$length;                             # remember this
  #print "border: $border length $length $mod\n";

  # now we add two arrays in a row, both are equal except the first element
  # which is in case A a step from the last inital prime to the second in list
  # and in case B a step from '1' to the second in list

  #print "add[border+1]: ",$add->[$border+1]," add[border] $add->[$border]\n";
  $wheel[$border] = $add->[$border+2]-$add->[$border];
  $wheel[$border+$length] = $add->[$border+2]-1;
  # and last add a wrap-around around $mod
  #print "last: ",$add->[-1],"\n";
  $wheel[$border+$length-1] = 1+$mod-$add->[-1];
  $wheel[$border+$length*2-1] = $wheel[$border+$length-1];
 
  $i = $border + 1;
  # now fill in the rest
  while ($i < $length+$border-1)
    {
    $wheel[$i] = $add->[$i+2]-$add->[$i+1];
    $wheel[$i+$length] = $wheel[$i];
    $i++;
    }
  ($ws,\@wheel);
  }

sub factors_wheel
  {
  my $n = shift;
  my $o = abs(shift || 1);

  $n = Math::BigInt->new($n) unless ref $n;
  my $two = Math::BigInt->new(2);
  my $three = Math::BigInt->new(3);

  my @factors = ();
  my $x = $n->copy();

  return ($x) if $x < 4;
  my ($i,$y,$w,$div,$rem);

  #print "Using a wheel of order $o, length ";
  my $wheel = wheel($o);
  #print scalar @$wheel,":\n";
  my ($ws,$add) = _transform_wheel($wheel); undef $wheel;
  my $we = scalar @$add - 1;

  # reduce to odd number (after that, no odd left-over divisior will ocur)
  while (($x->is_even) && (!$x->is_zero))
    {
    push @factors, $two->copy();
    #print "factoring $x (",$x->length(),")\n";
    #print "2\n";
    $x /= $two;
    }
  # 8 => 6 => 3, 7, 6 => 3, 5, 4 => 2 => 1, 3, 2 => 1, are all prime
  # so the first number interesting for us is 9
  my $op = 0;
 OUTER:
  while ($x > 8)
    {
    #print "factoring $x (",$x->length(),")\n";
    $i = $three; $w = 0;
    while ($i < $x)             # should be sqrt()
      {
      # $steps++;
      # $op = 0, print "$i\r" if $op++ == 1024;
      $y = $x->copy();
      ($div,$rem) = $y->bdiv($i);
      if ($rem == 0)
        {
        #print "$i\n";
        push @factors,$i;
        $x = $div; next OUTER;
        }
      #print "$i + ",$add->[$w]," ($w)\n";
      #$i += 2;                                   # trial div by odd numbers
      $i += $add->[$w];
      #print "restart $w $ws\n" if $w == $we;  # wheel of 2,3,5,7...
      $w = $ws if $w++ == $we;                  # wheel of 2,3,5,7...
      #exit if $i > 100000;
      }
    last;
    }
  push @factors,$x if $x != 1 || $n == 1;
  @factors;
  }

sub _factor
  {
  # later: factor ( n => $n, algorithmn => 'wheel', order => 3 );
  }

1;
__END__

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1;