Congruence Properties of Mersenne Primes

TL;DR

This paper investigates the properties of Mersenne primes, proving their form and exploring relationships with natural numbers, which could aid in understanding their distribution and cardinality.

Contribution

It establishes that all Mersenne primes are of the form 4n+3 and generalizes this to powers of 2, providing new insights into their structure.

Findings

Mersenne primes are of the form 4n+3

Generalization to all powers of 2

Discussion on methods to determine their cardinality

Abstract

In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer 'n'} and generalize the result to all powers of 2. We also tabulate and show their relationship with other whole numbers up to 10. A number of minor results are also proved. Based on these results, approaches to determine the cardinality of Mersenne primes are discussed.