Aurifeuillian factorizations and the period of the Bell numbers modulo a prime. (English) Zbl 0852.11008
The Bell integers \(B(n)\) are defined by \(\exp (\exp x-1)= \sum B(j) x^j/ j!\). The period mod \(p\) is considered for \(p\) prime [see G. T. Williams, Am. Math. Mon. 52, 323–327 (1945; Zbl 0060.08416)]. It is known to be a divisor of \(N_p= (p^p - 1)/(p - 1)\) (and conjecturally equal to \(N_p\)). This is verified for \(p < 102\) by eliminating all divisors of \(N_p\). The Aurifeuillian factorizations of \(N_p\) are algebraic, based on cyclotomy and serve as a starting point, for \(p\equiv 1\bmod 4\). (Factors of \(K_p (p^p + 1)/(p + 1)\) are included for good measure.) The inductive use of Bell identities, e.g., \(B(n+1)= \sum B(j) C^n_j\) and \(B(n+ p)= B(n)+ B(n+ 1)\) aid the calculation. Tables are included for factorizations by the elliptic curve and quadratic sieve for \(p < 180\).
Reviewer: Harvey Cohn (Bowie)
MSC:
| 11B73 | Bell and Stirling numbers |
| 11A51 | Factorization; primality |
| 11Y05 | Factorization |
| 12E10 | Special polynomials in general fields |
Citations:
Zbl 0060.08416Online Encyclopedia of Integer Sequences:
Table read by rows in which row n lists divisors of (p^p-1)/(p-1) where p = prime(n).References:
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