Infinite Primes, Quadratic Polynomials, and Fermat’s Criterion

EasyChair Preprint 11402

Abstract

In this study, we explore the existence of an infinite number of primes
represented by the quadratic polynomial 4(Mp − 2)2 + 1 . We propose
a hypothesis that considers Fermat primes as a criterion for the infinitude
of such primes, where Mp represents Mersenne primes. Additionally,
we provide an elementary argument supporting the presence of infinitely
many primes in the form , as these primes are a subset of primes of the
same form x2 + 1 . Furthermore, we present a basic argument demonstrating
the infinity of Mersenne primes. This paper contributes to the
understanding of prime numbers and their intriguing relationships with
quadratic polynomials and Fermat primes.

Keyphrases: Prime, Real, natural number

@booklet{EasyChair:11402,
  author    = {Budee U Zaman},
  title     = {Infinite Primes, Quadratic Polynomials, and Fermat’s Criterion},
  howpublished = {EasyChair Preprint 11402},
  year      = {EasyChair, 2023}}