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A family of super congruences involving multiple harmonic sums. (English) Zbl 1419.11003

Summary: In recent years, the congruence
\[ \sum_{\substack{i+j+k=p \\ i,j,k >0}} \frac{1}{ijk} \equiv -2 B_{p-3} \pmod p,\]
first discovered by the last author has been generalized by either increasing the number of indices and considering the corresponding super congruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar super congruences modulo prime powers \(p^r\) with the indices summing up to \(mp^r\) where \(m\) is coprime to \(p\), and where all the indices are also coprime to \(p\).

MSC:

11A07 Congruences; primitive roots; residue systems
11B68 Bernoulli and Euler numbers and polynomials
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References:

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