A curious congruence modulo prime powers (Q404320)
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scientific article; zbMATH DE number 6339615
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A curious congruence modulo prime powers |
scientific article; zbMATH DE number 6339615 |
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A curious congruence modulo prime powers (English)
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4 September 2014
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congruences
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Bernoulli numbers
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modulo prime powers
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Let \(B_n\) denote the \(n\)th Bernoulli number. The authors show that for any prime \(p\geq 3\) and positive integer \(r\) one has NEWLINE\[NEWLINE\sum_{i+j+k=p^r}{1\over ijk}\equiv -2p^{r-1}B_{p-3}\pmod{p^r},NEWLINE\]NEWLINE where \(i,j\) and \(k\) are positive integers that are coprime to \(p\). The case \(r=1\) was established earlier by \textit{J. Zhao} [J. Number Theory 123, No. 1, 18--26 (2007; Zbl 1115.11005)].
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