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).
Full version: pdf,
(Concerned with sequences
Received September 11 2020; revised version received February 26 2021.
Published in Journal of Integer Sequences,
April 25 2021.
Journal of Integer Sequences home page