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

Number Theory 2025-01-07 v2

Abstract

The Goormaghtigh conjecture states that the only two numbers which have two non-trivial representations as repunits are 3131 and 81918191. We call such a prime number a {\it Goormaghtigh prime}. We show that there are no other Goormaghtigh primes less than 1070010^{700}.

Keywords

Cite

@article{arxiv.2410.03677,
  title  = {No new Goormaghtigh primes up to $10^{700}$},
  author = {Jon Grantham},
  journal= {arXiv preprint arXiv:2410.03677},
  year   = {2025}
}

Comments

v2; published version + 1 typo corrected