Good observation! It looks like Fermat's method (the method I mentioned above) doesn't necessarily find all, or even any of the factors. It is just a heuristic. On the other hand, if you have an even number, you may extract factors of 2 until you get an odd number, in which case (x+y) and (x-y) must both be necessarily odd.

The generalization that gumby mentioned, x^2 = y^2 (mod p) may have more sucess, but I don't know if it is exhaustive. Here is code that implements it:

my $p = shift;
my %residues;
my %factors;
foreach my $x (2..$p) {
   my $mod = ($x*$x) % $p;
   if (exists $residues{$mod}) {
      $factors{$x-$residues{$mod}}++ if $p % ($x-$residues{$mod}) == 0
+;
      $factors{$x+$residues{$mod}}++ if $p % ($x+$residues{$mod}) == 0
+;
   }
   else {
      $residues{$mod} = $x;
   }
}
foreach (sort {$a <=> $b} keys %factors) {
   print "$p = $_*", $p/$_, "\n";
}
[download]

For 54, this yields

1021% factor.pl 54
54 = 2*27
54 = 6*9
54 = 18*3
54 = 54*1
[download]

But this code is unsatisfying because it takes longer than the simple trial by division. Thinking from a divide and conquer point of view, it is probably faster to use these methods to find a factorization, then recursively find factorizations of the factors untill all are prime.

-Mark

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