Maybe one of the mathematician's will show me the error of my ways, but I think that the prime factors thing is a red-herring.

The following brute force approach works quite quickly for numbers up to about 12 or 13 digits, usually much more quickly that it takes M::B::F to find the prime factors.

If you're looking to do this with very large numbers (as suggested by your use of Math::Big), then your probably in for a long wait either way. If the number in question is itself prime, then the nearest factor to the square root will be 1. That means you have to add 1 to list that factor_wheel() produces before you start doing your combinatorics, and it's presence just slows the search to what effectively amounts to a linear search anyway.

#! perl -slw
use strict;

my $NUM = $ARGV[ 0 ] || die 'No arg';

my $root = sqrt( $NUM );
my( $lo, $hi ) = ( int( $root ), int( $root + 1 ) );;

$lo-- while $NUM % $lo;
$hi++ while $NUM % $hi;

my $near = ( $lo, $hi )[ abs( $root - $lo ) > abs( $root - $hi ) ];

print "$NUM ($root) : $near";
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