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

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 $10^{700}$.

v2; published version + 1 typo corrected