I moved away from using M::B::F. The "best" I came up with is:

#! perl -slw
use strict;

my $NUM = $ARGV[ 0 ] || die 'No arg';
$NUM = eval $NUM if $NUM =~ m[\D];

my $root = sqrt( $NUM );
my $near = int $root;
my $step = $NUM % 2 ? 2 : 1;

$near-= $step while $near and  $NUM % $near;
$near = 1 unless $near;

printf "%10.f (%10f) %.f\n", $NUM, $root, $near;
[download]

It's a simple brute force search for a factor < root.

There may be a way to use the prime factors to speed the search, but the time they take to produce, a linear search down from the root is quicker.

Even with a largish prime like 988041964007, which means a linear search all the way to 1, this takes less than a second, where Math::Big::Factors::Factors_wheel() churns for hours trying to factorise it.

Maybe there's a qucker factorising module out there somewhere? Even so, from what I've tried and seen from other peoples attempts, having the prime factors doesn't give you any obvious way to avoid what amounts to a linear search.