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Prime divisors of the Bernoulli and Euler numbers. (English) Zbl 1050.11021
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 (ISBN 1-56881-152-7/hbk). 357-374 (2002).

The Bernoulli numbers may be defined by

x e x -1= n=0 B n x n n!·

It is known that (i) B 2n+1 =0 for all n1; (ii) the Bernoulli numbers are rational, that is, B 2n =N 2n /D 2n where N 2n ,D 2n for all n0; (iii) D 2n ={p:p prime (p-1)|2n}.

The Euler numbers may be defined by

secx= n=0 (-1) n E 2n x 2n (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 602k132, and of E 2k for 2k88. These new results extend some of his earlier efforts.


MSC:
11B68Bernoulli and Euler numbers and polynomials