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Zbl 1050.11021
Wagstaff, Samuel S. jun.
Prime divisors of the Bernoulli and Euler numbers. (English)
[A] Bennett, M. A. (ed.) et al., Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21--26, 2000. Natick, MA: A K Peters. 357-374 (2002). ISBN 1-56881-152-7/hbk

The Bernoulli numbers may be defined by $${x\over e^x- 1}= \sum^\infty_{n=0} {B_nx^n\over n!}.$$ It is known that (i) $B_{2n+1}= 0$ for all $n\ge 1$; (ii) the Bernoulli numbers are rational, that is, $B_{2n}= N_{2n}/D_{2n}$ where $N_{2n},D_{2n}\in\bbfZ$ for all $n\ge 0$; (iii) $D_{2n}= \prod\{p:p$ prime $(p-1)\vert 2n\}$.\par The Euler numbers may be defined by $$\text{sec\,}x= \sum^\infty_{n=0} (-1)^n E_{2n} {x^{2n}\over (2n)!}.$$ Prime factors of the Euler numbers, and of the numerators of the Bernoulli numbers, arise in the study of cyclotomic fields. The author obtains prime factorizations of $N_{2k}$ for $60\le 2k\le 132$, and of $E_{2k}$ for $2k\le 88$. These new results extend some of his earlier efforts.
[N. Robbins (San Francisco)]

MSC 2000:: *11B68 Bernoulli numbers, etc.

Keywords: Bernoulli numbers; Euler numbers

Cited in: Zbl 1204.11051

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