Only finitely many $s$-Cullen numbers are repunits for a fixed $s\ge 2$

Abstract

We show that for any integer $s\ge 2$, there are only finitely many $s$-Cullen numbers that are repunits. More precisely, for fixed $s\ge 2$, there are only finitely many integers $n$, $b$, and $q$ with $n\ge 2$, $b\ge 2$ and $q\ge 3$ such that \[C_{n,s} = ns^n + 1 = \frac{b^q -1}{b-1}.\] The proof is elementary and effective, and it is used to show that there are no $s$-Cullen repunits, other than explicitly known ones, for all $s\in [2,8896]$.

v2. Corrections and clarifications in computation section