| According to Philip Davis, "One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories" (Guillemets). This thesis examines some of the exciting historical developments surrounding one of these thorny dilemmas: Fermat's conjecture that numbers of the form 22n +1 are prime for all n∈Znonnegative . While it took almost 100 years before Euler found a counterexample, and another sixty years for Gauss to make the discovery that ignited interest in these numbers, serious mathematical strides resulting in beautiful theories have been made ever since. After examining several highlights pertaining to Fermat numbers, this paper focuses on three applications whose proofs rely heavily upon the power of Fermat primes: Gauss' aforementioned theorem followed by modern results due to Dr. Florian Luca (concerning Heron triangles) and Carrie Finch and Lenny Jones (regarding finite minimal POS groups). Lastly, the reader is offered several open problems. |
