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's elliptic curve test, before moving on to look at the Lenstra-Pomerance-Wagstaff conjecture, concerning the distribution of Mersenne numbers.
Advisor
Bush, Michael
Department
Mathematics
Recommended Citation
Robeson, Tanner, "A Treatise on the Theory of Mersenne Numbers and Primality Testing" (2025). Senior Independent Study Theses. Paper 11463.
https://openworks.wooster.edu/independentstudy/11463
Disciplines
Algebra | Analysis | Number Theory | Probability | Theory and Algorithms
Keywords
Mersenne numbers, Mersenne primes, primality test, primality testing, elliptic curve, number theory
Publication Date
2025
Degree Granted
Bachelor of Arts
Document Type
Senior Independent Study Thesis
© Copyright 2025 Tanner Robeson