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

Copyright © 1997 Paul Milnes and C. Stanley-Albarda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.