%I #28 Mar 07 2024 02:23:00
%S 1322314049613223140496,2066115702520661157025,2975206611629752066116,
%T 4049586776940495867769,5289256198452892561984,6694214876166942148761,
%U 8264462810082644628100,183673469387755102041183673469387755102041
%N Biperiod squares: square numbers whose digits repeat twice in order.
%D Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.
%D R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (1981-82), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (1982-83), 311-312.
%H Author?, <a href="http://www.math2.org/mmb/thread/31820">MMB message board "big square"</a>
%H Dr Barker, <a href="https://www.youtube.com/watch?v=c1peEN5Q-0c">Can Numbers Like These Be Square?</a>, YouTube video, 2023.
%H Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1707.03894">The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations</a>, preprint arXiv:1707.03894 [math.NT], July 14 2017.
%p f:=proc(n) local i,j,k; i:=cat(n,n); j:=convert(i,decimal,10); issqr(j); end;
%p with(numtheory): Digits:=50:for d from 1 to 22 do tendp1:=10^d+1: tendp1fact:=ifactors(tendp1)[2]: n:=mul(piecewise(tendp1fact[i][2] mod 2=1,tendp1fact[i][1],1),i=1..nops(tendp1fact)):for i from ceil(sqrt((10^(d-1))/n)) to floor(sqrt((10^d-1)/n)) do printf("%d, ",tendp1*n*i^2) od: od: # C. Ronaldo
%o (Python)
%o from itertools import count, islice
%o from sympy import sqrt_mod
%o def A092118_gen(): # generator of terms
%o for j in count(0):
%o b = 10**j
%o a = b*10+1
%o ab, aa = a*b, a*(a-1)
%o for k in sorted(sqrt_mod(0,a,all_roots=True)):
%o if ab <= (m:=k**2) < aa:
%o yield m
%o A092118_list = list(islice(A092118_gen(),10)) # _Chai Wah Wu_, Mar 06 2024
%Y Cf. A102567, A106497.
%K base,nonn
%O 1,1
%A Michael Mark, Dec 15 2004
%E Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005
%E Definition corrected and improved, reference and cross-reference added by _William Rex Marshall_, Nov 12 2010
%E Keyword base added by _William Rex Marshall_, Nov 12 2010