Congruences for Bernoulli numbers and Bernoulli polynomials
The Bernoulli numbers and polynomials are defined by \(B_0=1\), \(\sum^{n-1}_{k=0}{n\choose k} B_k= 0\) \((n=2,3,\dots)\) and \(B_n(x)= \sum^n_{k=0}{n\choose k} B_{n-k} x^k\), respectively. Two basic congruences for Bernoulli numbers are the Kummer congruences (used in the theory of Fermat's last theorem) and the von Staudt-Clausen theorem. There exist many extensions and generalizations for these two theorems. The author obtains a new generalization of the von Staudt-Clausen theorem, by proving that \[ \begin{multlined} pB_{k(p-1)+b}(x)- p^{k(p-1)+b} B_{k(p-1)+b}\Biggl({x+\langle-x\rangle\over p}\Biggr)\equiv\\ a_nk^n+ a_{n-1}k^{n-1}+\cdots+ a_1k+a_0\pmod{p^n}\quad (k=0,1,\dots),\end{multlined} \] where \(p\) is a prime; \(b\in\mathbb{Z}^+\cup\{0\}\); \(x\in\mathbb{Z}_p\); \(\langle-x\rangle\) is the least nonnegative residue of \(-x \text{mod }p\); and \(a_0,a_1,\dots,a_n\) are all (explicit) integers. For the reader's convenience we quote the excellent work by \textit{K. Dilcher}, \textit{L. Skula} and \textit{I. Sh. Slavutskij} [Bernoulli numbers. Bibliography (1713-1990). (1991; Zbl 0741.11001)].
scientific article; zbMATH DE number 1018625
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