Journal of Integer Sequences, Vol. 24 (2021), Article 21.5.3

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Jon Grantham and Hester Graves
Institute for Defense Analyses
Center for Computing Sciences
Bowie, MD 20715


Assuming the abc conjecture holds with ε = 1/6, we use elementary methods to show that only finitely many s-Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers s, n, b, and q with s,b ≥ 2 and n,q ≥ 3 satisfy Cs,n = nsn + 1 = (bq–1)/(b–1).

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(Concerned with sequences A000225 A001110 A002064 A002275 A050914 A135518.)

Received September 11 2020; revised version received February 26 2021. Published in Journal of Integer Sequences, April 25 2021.

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