The other side of Mersenne Primes

Mersenne numbers, (2^n)-1, with natural number n, are like pay dirt, containing dirt and apparent gold. They entice us to ask: which of these numbers are prime, which are not, and if not, why not - especially the shiny ones? We address the "why not?" of that question here, exploring the nature of all Mersenne composites, the other side of Mersenne primes. There are two main themes to this work: explorations into Mersenne numbers, and examples of community engagement with the concepts involved. Chapter 1 presents Mersenne numbers visually, using modular arithmetic clocks. Chapter 2 digs at the scope and cause of Mersenne composite construction. We take a Sieve of Eratosthenes approach, customized to Mersenne numbers. Here, we see how metaphorical fool's gold - Mersenne candidates that look prime but are composite - occur, through a behaviour we coin telescoping, and we examine the nature of telescoping. Chapter 3 presents the famed Lucas-Lehmer primality test, along with proofs. Chapter 4 presents a collection of enrichment modules, catered to highschoolers, maths enthusiasts, and crafters, that I have developed during my study. Chapter 5 closes with thoughts on further explorations, and the Appendices house Maple code and graphical data. In the spirit of community engagement, this thesis is written with the mathematically curious in mind, as an open invitation for research and play.