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A290172
Bases b for which there exists an integer y such that y^2 in base b consists of three identical digits.
5
18, 22, 30, 68, 146, 292, 313, 423, 439, 499, 521, 581, 653, 699, 710, 787, 1047, 1353, 1425, 1660, 1714, 2060, 2174, 2198, 2272, 2819, 3019, 3130, 3445, 3789, 4366, 4526, 4611, 4620, 4624, 4701, 4788, 4972, 5261, 5421, 5656, 6057, 6106, 6158, 6205, 6895, 6927, 7163, 7527, 7627, 7733, 9317, 9353, 10761, 11092
OFFSET
1,1
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.
LINKS
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.
EXAMPLE
For example, for b = 18, we have y = 49, and the base-b representation of y^2 is 777.
MATHEMATICA
r[b_] := Reduce[0 < x < b && y > 0 && y^2 == x + b x + b^2 x, {x, y}, Integers]; Reap[For[b = 2, b < 12000, b++, If[r[b] =!= False, Print[b]; Sow[b]]]][[2, 1]] (* Jean-François Alcover, Jul 23 2017 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, Jul 23 2017
STATUS
approved