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 $N_p = (p^p-1)/(p-1)$. It is known that the period always divides $N_p$. The period is shown to equal $N_p$ for most primes $p$ below 180. The investigation leads to interesting new results about the possible prime factors of $N_p$. 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 $p^{m^2p}-1$. Then we explain how this theorem influences the probability that $q$ divides $N_p$.