Almost prime

Prime power

Prime k-tuple

Discrete mathematics

Prime (order theory)

Computer science

Acoustics

Integer (computer science)

Radical of an integer

Divisor (algebraic geometry)

Prime factor

Physics

Mathematics

Prime number

Combinatorics

Programming language

Modulo

Period (music)

Primitive root modulo n

Arithmetic

Mathematics of Computation

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 $N_p$.

The period of the Bell numbers modulo a prime