Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun

Details Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun

2011-08-04

arxiv.org/abs/1108.1171v2

Journal-ref: Int. J. Number Theory, Vol. 8, No. 4(2012), 5 pages

Mathematics

Number Theory

This is a final version accepted for publication in International Journal of Number Theory

Scientific paper

10.1142/S1793042112500649

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic
number. In this note we prove that for any prime $p\ge 7$,
$$ \sum_{k=1}^{p-1}\frac{H_k^2}{k^2} \equiv4/5pB_{p-5}\pmod{p^2},
$$ which confirms the conjecture recently proposed by Z. W. Sun. Furthermore,
we also prove two similar congruences modulo $p^2$.

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