The period of the Bell numbers modulo a prime

Abstract

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (p P — 1)/(p - 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q = 4m 2 p + 1 is prime, then q divides p m2p - 1. Then we explain how this theorem influences the probability that q divides Np.