On some congruences using multiple harmonic sums of length three and four
In the present paper, we determine the sums $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}}{j^{s_2}}$ and $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}H_j^{(s_4)}}{j^{s_2}}$ modulo $p$ and modulo $p^2$ in certain cases. This is done by using multiple harmonic sums of length three and four, as well as, many other results. In addition, We recover three congruences conjectured by Z.-W Sun and solved later by the author himself and R. Meštrović.
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