International Journal of Mathematics and Mathematical Sciences 
Volume 20 (1997), Issue 3, Pages 521-528
doi:10.1155/S0161171297000719

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

Paul Milnes1 and C. Stanley-Albarda2

 1Department of Mathematics, University of Western Ontario, Ontario, London N6A 5B7, Canada
2Department of Mathematics, University of Toronto, Ontario, Toronto M5S 1A1, Canada
Received 6 November 1995

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.