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The period of the Bell numbers modulo a prime

Authors: Peter L. Montgomery, Sangil Nahm and Samuel S. Wagstaff Jr.
Journal: Math. Comp. 79 (2010), 1793-1800
MSC (2010): Primary 11B73, 11A05, 11A07, 11A51
Published electronically: March 1, 2010
MathSciNet review: 2630013
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Abstract | References | Similar Articles | Additional Information

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$.

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Additional Information

Peter L. Montgomery
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052

Sangil Nahm
Affiliation: Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907-2067

Samuel S. Wagstaff Jr.
Affiliation: Center for Education and Research in Information Assurance and Security, and Departments of Computer Science and Mathematics, Purdue University, 305 North University Street, West Lafayette, Indiana 47907-2107

Keywords: Bell numbers, period modulo $p$
Received by editor(s): July 9, 2008
Received by editor(s) in revised form: August 7, 2009
Published electronically: March 1, 2010
Additional Notes: This work was supported in part by the CERIAS Center at Purdue University.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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