Mathematics Seminar Talk - Pete Clark

A New Proof of the Generalized Wilson’s Theorem

Wilson’s Theorem is a staple of elementary number theory: for a prime number p, (p-1)! is congruent to -1 modulo p. Suppose we would like to generalize Wilson’s theorem. Why? Well, the title says so: in fact it is, verbatim, the title of a 1903 paper by the American group theorist G.A. Miller. But more fundamentally: why not? Shouldn’t we be trying to generalize all the theorems we meet? I think so. In this talk we will contemplate several possible generalizations and then hit upon two that are discussed in Miller’s paper. We will prove one of these by a method which is more elementary than Miller’s (i.e., a new proof).