Newsgroups: sci.math
From: moeller@gwdgv1.gwdg.de
Subject: puzzling squares
Date: Thu, 8 Oct 1992 00:29:37 GMT

Have you ever seen a (rational) square number which,
when written in decimal notation, has the same sequence of digits
left and right of the decimal point?

To my surprise, I noticed that such numbers do exist, but are *large*.
The smallest one I know is the square of

	((10^136 + 1)/17) * 6 * 10^(-68)

To prove that this is indeed the smallest such number,
one had to show that  (10^(2*k) + 1)  is square-free, for (2*k) < 136.
Does the current state of the "art of factorization" allow for this check?

Wolfgang J. Moeller, GWDG, D-3400 Goettingen, F.R.Germany | Disclaimer ...
PSI%(0262)45050352008::MOELLER      Phone: +49 551 201516 | No claim intended!
Internet: moeller@gwdgv1.dnet.gwdg.de   | This space intentionally left blank.
==============================================================================

Date: Fri, 6 Feb 1998 14:59:16 +1100
To: rusin@math.niu.edu
From: Gerry Myerson <gerry@mpce.mq.edu.au>

[deletia]

I think the factorizations necessary can be found in Brillhart et al.,
Factorizations of b^n\pm1, 2nd ed., Amer Math Soc 1988.   136 is the
smallest value of 2k for which 10^{2k} + 1 has what the authors call
"an intrinsic prime factor."

The only other such value listed is 202 which has the intrinsic prime 
factor 101; presumably this could be used to concoct another example of the
A.A = square phenomenon.