The original question started "given N and N's prime factorization", so it seems inappropriate to take into account the time taken to factorize.

In this case though, note that 988041964007 = 7 x 11087 x 12731023 - you are losing accuracy by throwing out the BigInts.

In general if the factorization of N is taking longer than it takes to do sqrt(N) trial divisions then the factorization algorithm is pretty bad, since "trial division up to the square root" is the classical example of a slow algorithm.

Many of the high precision maths libraries out there do factorization - for example the pari library (Math::Pari) has a factorint() function:

  use Math::Pari qw/ factorint /;
  my $n = Math::Pari->new("988041964007");
  my $f = factorint($n);
  my $count = $#{ $f->[0] };
  print join(" * ", map {
    my($prime, $power) = ($f->[0][$_], $f->[1][$_]);
    $power > 1 ? "$prime ^ $power" : $prime
  } 0 .. $count), "\n";
7 * 11087 * 12731023

[download]

Note that this algorithm does not guarantee primality: for large factors you need to verify with an isprime() check.

Hugo

In reply to Re^6: OT: Finding Factor Closest To Square Root by hv
in thread OT: Finding Factor Closest To Square Root by QM

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