Analogues of the Binomial Coefficient Theorems of Gauss and Jacobi

Abstract

Two of the more well known congruences for binomial coefficients modulo p, due to Gauss and Jacobi, are related to the representation of an odd prime (or an integer multiple of the odd prime) p as a sum of two squares (or an integer linear combination of two squares). These two congruences, along with many others, have been extended to analogues modulo p^2 and are well documented. More recently, J. Cosgrave and K. Dilcher have extended the congruences of Gauss and Jacobi to analogues modulo p^3. In this thesis we discuss their methods as well as the potential of applying them to similar congruences found in the literature.