The proof of the Primitive Divisor Theorem

Abstract

This dissertation provides the main results leading to its primary aim, the proof of the Primitive Divisor Theorem, by appealing to an electric potpourri of mathematical machinery. The employment of binary recurrent sequences with related results is crucial to the approach adopted. The various forms in which the theorem manifests are attributed, among others, to K. Zsigmondy, P.D. Carmichael, and Y. Bilu, G. Hanrot and P.M. Voutier. The proof is confined to instances where the roots of the characteristic polynomial are integers, and when the roots are reals. This dissertation culminates in the resolution of a Diophantine equation which serves as an application of the Primitive Divisor Theorem that is attributed to Carmichael.