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

TL;DR

Under the assumption of the abc conjecture, the paper proves that only finitely many s-Cullen numbers can be repunits, except for two known infinite families, using elementary methods.

Contribution

It establishes a finiteness result for s-Cullen numbers that are repunits, assuming the abc conjecture, and clarifies the rarity of such numbers.

Findings

Finitely many s-Cullen numbers are repunits under abc conjecture

Two known infinite families of such numbers are exceptions

Elementary methods suffice for the proof

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 $s,b \geq 2$ and $n,q \geq 3$ satisfy \[C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.\]