Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun (Q2890257)

This paper proves that for any prime \(p\geq 7\), NEWLINE\[NEWLINE \sum_{k=1}^{p-1} \frac{H_k^2}{k^2}\equiv \frac 45 pB_{p-5}\pmod {p^2}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \sum_{k=1}^{p-1} \frac{H_k^3}{k}\equiv \frac 32 pB_{p-5}\pmod {p^2}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \sum_{k=1}^{p-1} \frac{H_k}{k^3}\equiv \frac {-1}{10} pB_{p-5}\pmod {p^2}, NEWLINE\]NEWLINE being \(B_0, B_1, B_2,\dots\), the Bernoulli numbers and \(H_n=\sum_{k=1}^n 1/k\) the \(n\)th harmonic number for a positive integer \(n\) (with \(H_0=0\)).NEWLINENEWLINEThe first congruence is the Conjecture \(A37\) proposed by \textit{Zhi-Wei Sun} [``Open conjectures on congruences'', arXiv:0911.5665v54] whose remarkable work [Proc. Am. Math. Soc. 140, No. 2, 415-428 (2012; Zbl 1271.11021)], generalized by \textit{R. Tauraso} [Sémin. Lothar. Comb. 63, B63g, 8 p., (2010; Zbl 1273.11007)] and further refined by the same \textit{Z.-W. Sun} and \textit{L.-L. Zhao} [Colloq. Math. 130, No. 1, 67--78 (2013; Zbl 1290.11052)], has motivated the author to find not only this proof but also an interesting similar one \textit{Romeo Meštrović} [``On the mod \(p^2\) determination of \(\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)\): another proof of a conjecture by Sun'', \url{arXiv:1108.3197}].NEWLINENEWLINEHere the author uses the Wolstenholme's theorem studied by \textit{A. Granville} [Borwein, J. (ed.) et al., Organic mathematics. Proceedings of the workshop, Simon Fraser University, Burnaby, Canada, December 12-14, 1995. Providence, RI: American Mathematical Society. CMS Conf. Proc. 20, 253--276 (1997; Zbl 0903.11005)], by \textit{E. Alkan} [Am. Math. Mon. 101, No. 10, 1001--1004 (1994; Zbl 0836.11002)], by \textit{M. Bayat} [Am. Math. Mon. 104, No. 6, 557--560 (1997; Zbl 0916.11002)] and by \textit{J. Zhao} [Int. J. Number Theory 4, No. 1, 73--106 (2008; Zbl 1218.11005)] who also established some congruences for multiple harmonic numbers essential to complete the proof; an identical result from \textit{X. Zhou} and \textit{T. Cai} [Proc. Am. Math. Soc. 135, No. 5, 1329--1333 (2007; Zbl 1115.11006)] is recalled too.