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

TL;DR

The paper proves that for any fixed integer s ≥ 2, only finitely many s-Cullen numbers can be repunits, and explicitly confirms this for all s in the range [2, 8896], using an elementary and effective proof.

Contribution

It establishes finiteness of s-Cullen numbers that are repunits for fixed s and verifies this for a large range of s through an elementary proof.

Findings

Finiteness of s-Cullen repunits for fixed s ≥ 2.

No new s-Cullen repunits found for s in [2, 8896].

Elementary and effective proof method used.

Abstract

We show that for any integer $s \geq 2$, 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 $n \geq 2$, $b \geq 2$ and $q \geq 3$ 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 $s \in [2,8896]$.