No new Goormaghtigh primes up to $10^{700}$
Jon Grantham
TL;DR
The work addresses the Goormaghtigh prime problem, seeking primes $N$ that have two repunit representations $N=\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}$ with $y>x\ge 2$ and $n>m>2$. It leverages structure theorems (from Bennett–Garbuz–Martens and related results) to constrain $m,n$ (notably $m,n$ being odd primes with $m\ge 53$ and excluding many gcd cases) and employs a modular-pruning algorithm on $f_k(x)=\frac{x^k-1}{x-1}$ across small moduli to exhaustively search up to $10^{700}$. The main finding is that no new Goormaghtigh primes arise below $10^{700}$, even when requiring both exponents to be prime, and the work provides a slightly stronger statement about the absence of new primes under these bounds. A conditional result based on the abc conjecture suggests there are finitely many Goormaghtigh numbers (and hence primes) in the absence of small-length representations, reinforcing the rarity of such primes and guiding future computational and theoretical effort.
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 $10^{700}$.