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 [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.