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), 17931800
MSC (2010):
Primary 11B73, 11A05, 11A07, 11A51
Published electronically:
March 1, 2010
MathSciNet review:
2630013
Fulltext PDF
Abstract 
References 
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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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A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp., 16:363367, 1962. MR 0148632 (26:6139)
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 C. E. Bickmore.
Problem 13058. Math. Quest. Educ. Times, 65:78, 1896.
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 M. Car, L. H. Gallardo, O. Rahavandrainy, and L. N. Vaserstein.
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The least prime in certain arithmetic progressions. Amer. Math. Monthly, 116:641643, 2009.
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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 479072067
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 479072107
Email:
ssw@cerias.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0025571810023409
PII:
S 00255718(10)023409
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.
