Undergraduate Honors Thesis Projects

Date of Award

4-2020

Document Type

Honors Paper

Degree Name

Mathematics-BS

Department

Mathematical Sciences

Advisor

Jeremy Moore, Ph.D.

First Committee Member

Jeremy Moore, Ph.D.

Second Committee Member

Ryan Berndt, Ph.D.

Third Committee Member

Halard Lescinsky, Ph.D.

Keywords

Mathematics, Prime numbers, Mersenne Primes, Perfect Numbers, Number Theory

Subject Categories

Number Theory

Abstract

The prime numbers have been an important field of research for thousands of years and are intertwined with most other fields of mathematics. One topic that has piqued the interest of mathematicians young and old is the Mersenne prime numbers, which have applications in many mathematics and computer science fields. The Mersenne primes get a lot of attention because there is not much known about them. However, we do have a very simple primality test for Mersenne numbers, which is why the largest currently known primes are Mersenne primes. These primes are also very closely related to another class of numbers called the perfect numbers. In fact, every even perfect number has an underlying Mersenne prime, and for every Mersenne prime we can find an even perfect number. However there is still a major question that remains unanswered: Are there an infinite number of Mersenne primes? In this paper we outline the various known results regarding Mersenne prime numbers and how each of these results helps us to move closer to finding the answer to this age old question.

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