[seqfan] Palindromic Squares

[Hans Havermann]

https://oeis.org/A002779 and their roots https://oeis.org/A002778 currently have b-files of 485 terms uploaded by Tony Noe, attributed to Patrick De Geest, although it's safe to assume that Patrick got them from Feng Yuan whose 2001/2002 calculations are still http://www.fengyuan.com/palindrome.html on the web. Feng's compilation is missing palindromic squares of lengths 46-48 and ten of the final eleven entries for length-49 are either corrupted or outright wrong. (One website reproduced the final entry without apparently realizing this.)

Collating Feng's list to the end of length-45 yields 1940 terms (the breakdown by palindrome digit-length is 4, 0, 3, 0, 7, 1, 5, 0, 11, 0, 5, 1, 19, 0, 13, 1, 25, 0, 18, 0, 48, 1, 31, 0, 70, 1, 44, 2, 105, 0, 70, 1, 153, 1, 98, 3, 209, 0, 132, 0, 291, 1, 181, 1, 384), including 61 prime https://oeis.org/A065378 examples.

I'm having difficulty understanding how Feng achieved his computations and therefore reproducing them. For example, he has three solutions for 36-digit palindromic squares 404099764753665981^2, 633856150760638652^2, and 795559265009384106^2, achieved (he states) in under 3 seconds. Could someone enlighten me with an efficient programming approach?