I went from the other direction: Look at only the squares in the range: 
use strict;
use warnings;
use POSIX 'ceil';

my $digits = 6;
my $fd = $digits/2;
my ($lo, $hi) = ((9 x ($digits-1))+1, 9 x $digits);
for my $sqrt (ceil(sqrt $lo) .. int(sqrt $hi)) {
  my $prod = $sqrt * $sqrt;
  next if substr($prod, $fd, 1) eq '0';
  my ($f1, $f2) = (substr($prod, 0, $fd), substr($prod, $fd));
  print "$f1 + $f2 = $sqrt, the root of $prod\n" if $f1 + $f2 == $sqrt
+;
}

[download]
Caution: Contents may have been coded under pressure.

In reply to Re^2: Vampire Numbers Revisited by Roy Johnson
in thread Vampire Numbers Revisited by Limbic~Region

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.