A092118 Biperiod squares: square numbers whose digits repeat twice in order.

1322314049613223140496, 2066115702520661157025, 2975206611629752066116, 4049586776940495867769, 5289256198452892561984, 6694214876166942148761, 8264462810082644628100, 183673469387755102041183673469387755102041

Offset: 1

Author

Michael Mark, Dec 15 2004

Keywords

• base
• nonn

References

• 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.
• 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.

Links

• Author?, MMB message board "big square"
• Dr Barker, Can Numbers Like These Be Square?, YouTube video, 2023.
• Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017.

Crossrefs

Cf. A102567, A106497.

Programs

Maple

f:=proc(n) local i,j,k; i:=cat(n,n); j:=convert(i,decimal,10); issqr(j); end;
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

Python

from itertools import count, islice
from sympy import sqrt_mod
def A092118_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        ab, aa = a*b, a*(a-1)
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if ab <= (m:=k**2) < aa:
                yield m
A092118_list = list(islice(A092118_gen(),10)) # Chai Wah Wu, Mar 06 2024

Extensions

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005

Definition corrected and improved, reference and cross-reference added by William Rex Marshall, Nov 12 2010

Keyword base added by William Rex Marshall, Nov 12 2010