A Treatise on the Theory of Mersenne Numbers and Primality Testing

Abstract

This thesis explores Mersenne numbers, numbers of the form $2^p-1$ where $p$ is prime. We are particularly concerned with when such numbers are themselves prime. We proceed rigorously and along a relatively consistent historical timeline, beginning with theory developed by Euclid in around 300 BCE and continuing through recent conjectures made in the late 20th century, as well as some elliptic curve theory. We start with some basic number theory and introduce the theory of quadratic residues to show that the prime factors of Mersenne numbers may only take certain forms. After that, we assume an algebraic approach to prove and discuss several primality tests, including the Lucas-Lehmer test, and Lenstra\u27s elliptic curve test, before moving on to look at the Lenstra-Pomerance-Wagstaff conjecture, concerning the distribution of Mersenne numbers