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On some congruences using multiple harmonic sums of length three and four (2004.12122v1)
Published 25 Apr 2020 in math.NT and math.CO
Abstract: 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 $p2$ 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. Me\v{s}trovi\'c.