Proof of a supercongruence conjectured by Z.-H. Sun

The Franel numbers are defined by $f_n=\sum_{k=0}^n {n\choose k}^3.$ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv 0\pmod{n{2n\choose n}}, \\ \sum_{k=0}^{p-1}\frac{3k+1}{(-16)^k} {2k\choose k} f_k &\equiv p (-1)^{\frac{p-1}{2}} \pmod{p^3}. \end{align*} where $n>1$ and $p$ is an odd prime. The second congruence modulo $p^2$ confirms a recent conjecture of Z.-H. Sun. We also show that, if $p$ is a prime of the form $4k+3$, then $\sum_{k=0}^{p-1}\frac{{2k\choose k} f_k}{(-16)^k} \equiv 0 \pmod p,$ which confirms a special case of another conjecture of Z.-H. Sun.

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