The period of the Bell numbers modulo a prime

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 = (pp - 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 = 4m2p + 1 is prime, then q divides pm2

• Type: Journal
• Year: 2010
• Where: MOC
• Authors: Peter L. Montgomery, Sangil Nahm, Samuel S. Wagstaff Jr.
• Keywords: MOC 2010, Odd Positive Integer, Positive Integer, Possible Prime Factors, Security Privacy