Generalizations of theorems of Wilson, Fermat and Euler

Abstract

By using Pólya's and de Bruijn's theorems of enumeration, we prove some generalizations of Wilson's, Fermat's and Euler's theorems in number theory. We also present an algorithm for Pólya's and de Bruijn's theorems. By using the properties of the algorithm, we interpret the meanings of the integers ( 1 p )((p − 1)! + 1) and ( 1 p )(a p − a) where p is a prime and a is a positive integer.