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

DOI:
https://doi.org/10.1090/S0025-5718-10-02340-9

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 can be a proper divisor of . It is known that the period always divides . The period is shown to equal for most primes below 180. The investigation leads to interesting new results about the possible prime factors of . For example, we show that if is an odd positive integer and is a positive integer and is prime, then divides . Then we explain how this theorem influences the probability that divides .

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Additional Information

**Peter L. Montgomery**

Affiliation:
Microsoft Research, One Microsoft Way, Redmond, Washington 98052

Email:
pmontgom@cwi.nl

**Sangil Nahm**

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

Email:
snahm@purdue.edu

**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

Email:
ssw@cerias.purdue.edu

DOI:
https://doi.org/10.1090/S0025-5718-10-02340-9

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.