No new Goormaghtigh primes up to $10^{700}$

TL;DR

This paper proves that no new Goormaghtigh primes exist below 10^700, confirming the conjecture's limited solutions within this range.

Contribution

The authors computationally verify the absence of Goormaghtigh primes below 10^700, extending the known bounds and supporting the conjecture.

Findings

No Goormaghtigh primes found below 10^700

Supports the conjecture's validity within this range

Extends previous computational verifications

Abstract

The Goormaghtigh conjecture states that the only two numbers which have two non-trivial representations as repunits are $31$ and $8191$. We call such a prime number a {\it Goormaghtigh prime}. We show that there are no other Goormaghtigh primes less than $10^{700}$.