On Some Generalizations of Fermat’s, Lucas’s and Wilson’s Theorems

Abstract

In \([1]\) and \([4]\), the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this lemma by applying Burnside’s theorem to the corresponding action by an arbitrary finite cyclic group. Although this idea is not new, by revisiting the constructions in \([1]\) and \([4]\) we derive three divisibility theorems for which the aforementioned classical theorems are, respectively, the cases of a prime divisor, and two of these generalizations are new. Throughout, \(n\) and \(p\) denote positive integers with \(p\) prime and \(\mathbb{Z}_n\) denotes the cyclic group of integers under addition modulo \(n\).