Congruences involving Bernoulli and Euler numbers

JOURNAL OF NUMBER THEORY

卷:128

Article

Sun, Zhi-Hong

关键词: Congruence;    Bernoulli number;    Bernoulli polynomial;    Euler number;   

DOI  :  10.1016/j.jnt.2007.03.003

来源: Elsevier

【 摘 要 】

Let [x] be the integral part of x. Let p > 5 be a prime. In the paper we mainly determine Sigma([p/4])(x=1) 1/x(k) (mod p(2)), ((p-1)([p/4]))(mod p(3)), Sigma(p-1)(k=1)2(k/)k (mod p(3)) and Sigma(p-1)(k=1)2(k)/k(2) (mod p(2)) in terms of Euler and Bernoulli numbers. For example, we have Sigma([p/4])(x=1)1/x(2) equivalent to (-1)(p-1/2)(8E(p-3) - 4E(2p-4)) + 14/3 pB(p-3) (mod p(2)), where E(n) is the nth Euler number and B(n) is the nth Bernoulli number. (C) 2007 Elsevier Inc. All rights reserved.

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