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

Abstract

Assuming the abc conjecture with $\epsilon =1/6$, 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 $s,b\ge 2$ and $n,q\ge 3$ satisfy \[C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.\]

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