Multiplicative polynomials and Fermat's little theorem for non-primes

Abstract

Fermat's Little Theorem states that xp=x(modp) for x∈N and prime p, and so identifies an integer-valued polynomial (IVP) gp(x)=(xp−x)/p. Presented here are IVP's gn for non-prime n that complete the sequence {gn|n∈N} in a natural way. Also presented are characterizations of the gn's and an indication of the ideas from topological dynamics and algebra that brought these matters to our attention.