The period of the Bell numbers modulo a prime
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- by Peter L. Montgomery, Sangil Nahm and Samuel S. Wagstaff, Jr.;
- Math. Comp. 79 (2010), 1793-1800
- DOI: https://doi.org/10.1090/S0025-5718-10-02340-9
- Published electronically: March 1, 2010
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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 $N_p$.References
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Bibliographic 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
- MR Author ID: 179915
- Email: ssw@cerias.purdue.edu
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2630013