Summary: 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 \( N_p\).