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 x 2 + 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 primesÂ