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Mathematics of Computation

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Aurifeuillian factorizations and the period of the Bell numbers modulo a prime

Author: Samuel S. Wagstaff Jr.
Journal: Math. Comp. 65 (1996), 383-391
MSC (1991): Primary 11--04, 11B73; Secondary 11Y05, 12--04, 12E10, 12Y05
MathSciNet review: 1325876
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Abstract: We show that the minimum period modulo $p$ of the Bell exponential integers is $(p^p-1)/(p-1)$ for all primes $p<102$ and several larger $p$. Our proof of this result requires the prime factorization of these periods. For some primes $p$ the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.

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Additional Information

Samuel S. Wagstaff Jr.
MR Author ID: 179915

Keywords: Bell numbers, period modulo $p$, integer factorization, Lucas’ identities, Aurifeuillian factorization
Received by editor(s): August 24, 1993
Received by editor(s) in revised form: January 26, 1995
Additional Notes: Some of the computing reported in this work was performed on a MasPar computer at Purdue University which was supported in part by NSF Infrastructure Grant CDA-9015696.
Article copyright: © Copyright 1996 American Mathematical Society