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Congruences for Bernoulli numbers and Bernoulli polynomials

✍ Scribed by Zhi-Hong Sun

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 292 KB
Volume: 163
Category: Article
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(97)81050-3

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✦ Synopsis

Let {B.(x)} be the well-known Bernoulli polynemials. It is the purpose of this paper to determine pB~p-t~+b(x)modp ", where p is a prime, k, b nonnegative integers and x a rational p-integer. It is interesting to investigate arithmetic properties of {B,} and {Bn(x)}. For the work on this line one may consult [1-3, 5-9, 11, 12]. Here we give two classical results (cf. [83): Kummer's congruences: Let p be an odd prime, and b an even number with p -1 Y b. For k = 0.1, 2 .... we have B~p-l)+b _ Bb (modp). k(p-l)+bb Von Staudt-Clausen Theorem: Suppose that p is a prime and k ¢ Z +. Then ~0(mod p) /f p ---2 and k > 1 is odd, pBk(p-1) --~ _ 1 (mod p) otherwise.

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