This is no great revelation post, merely a progress report. See the script below, which is an automation of the procedure I am using and analyzing so far. It’s not nearly the quickest – I expect extreme slowness for anything remotely near the RSA challenge number! – but it’s a starting point which I will modify upon discovery of ways to use the patterns created by the integer floors of the sqrt(Discriminant) values for the series of polynomials I am solving here.

#!/usr/bin/perl

use strict;
use warnings;
use POSIX qw/ceil/;
use POSIX qw/floor/;

my $m = $ARGV[0];
print “Factoring $m…\n”;
my $s = ceil( sqrt( $m ) );
my $r = $s * $s – $m;
my $b = 0;
my $c = 0 – $r;
my $disc = $b * $b – 4 * $c;
my $rdisc = sqrt( $disc );
my $frdisc = floor( $rdisc );
while ( $frdisc != $rdisc ) {
$b++;
$c -= $s;
$disc = $b * $b – 4 * $c;
$rdisc = sqrt( $disc );
$frdisc = floor( $rdisc );
}
print “Factorization found at n = $b, Root(D) = $rdisc.\n”;

print “p = “, $s + 0.5 * ( $b – $rdisc ), “; q = “, $s + 0.5 * ( $b + $rdisc );
print “\n”;