Integer Powers Expressed as Nested Sums of Lower Powers modulo 2, 3

Abstract

In this paper, we explore identities that allow for representation of positive integers raised to positive integral powers as sums of nested sums of smaller positive integral powers. We begin by establishing the base identity involving consecutive descending powers, which we then employ to construct a simple proof of Fermat's Little Theorem. After that, the focus shifts to deriving and rigorously proving a more intricate identity that represents integer powers as nested sums of descending powers of the same parity, i.e., descending powers modulo 2. The discussion is concluded by stating a prominent identity that allows for the representation of integer powers by lower powers modulo 3.