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Zbl 1024.11011
Slavutskii, Ilya
About von Staudt congruences for Bernoulli numbers. (English)
[J] Comment. Math. Univ. St. Pauli 48, No. 2, 137-144 (1999). ISSN 0010-258X

From the text: The author considers the von Staudt type congruences for Bernoulli numbers $B_n$ with arbitrary indices $n$ (the case $n\equiv 0$ $p\bmod{-1}$ being no exception). The theorems proved generalize well-known results due to H. S. Vandiver, L. Carlitz and others.\par Theorem 1. Let $p$ be an odd prime, $n=k(p-1)p^{l-1}$, $k$ and $l\in\bbfN$. Then $pB_n \equiv p-1\pmod{p^l}$ or more exactly $$pB_n\equiv p-1+kp^l\sum^{p-1}_{a=1} (a^{p-1} -1)/p\pmod {p^{l+1}},\ p>3.$$ Theorem 2. Let $k,l,t,u\in\bbfN$ and $z\in\bbfN\cup\{0\}$. Then $$\delta^{t+k-1}B^{hz+u}\equiv 0\pmod{p^{lt}},$$ where $h=\varphi(p^l)= (p-1)p^{l-1}$, $p^k\ge(2k+lt) (p-1)+1$ and $u\ge l(t+k-1)$. In particular, for $k=2$ we have $$\delta^{t+1} B^{hz+u}\equiv 0\pmod {p^{lt}},$$ or (in the usual symbolic form) $$B^{hz+u} (B^h-1)^{t+1}\equiv 0\pmod {p^{lt}},$$ where $p\ge lt+3$ and $u\ge l(t+1)$. $B_n$ denotes the Bernoulli number.

MSC 2000:: *11B68 Bernoulli numbers, etc.
11A07 Congruences, etc.

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