Primes of the Form 2x^2 - 1 and Their Connection to Near-Square Mersenne Primes

EasyChair Preprint 15543

Abstract

In this paper, we explore prime numbers of the form 2x^2 − 1 and investigate their distribution.
Through rigorous analysis, we demonstrate a high probability of the occurrence of such primes.
Furthermore, we present an argument for the infinitude of primes of this form, drawing on the
relationship between these primes and the near-square primes of Mersenne numbers. Our findings
provide a deeper understanding of the structure and properties of this class of primes, contributing
to broader research in number theory.

Keyphrases: Fermat prime, Mersenne primes, near-square primes, probability

@booklet{EasyChair:15543,
  author    = {Budee U Zaman},
  title     = {Primes of the Form 2x^2 - 1 and Their Connection to Near-Square Mersenne Primes},
  howpublished = {EasyChair Preprint 15543},
  year      = {EasyChair, 2024}}