Treffer: Proof of a supercongruence modulo p2r.

Employing Watson's terminating 8ϕ7$_8\phi _7$ transformation, we present a q$q$‐analog of the following supercongruence: for any prime p≡1(mod4)$p\equiv 1\pmod {4}$ and positive integer r$r$, ∑k=0pr−14k+164k2kk3≡0(modp2r),$$\begin{equation*} \sum _{k=0}^{p^{r}-1} \frac{4k+1}{64^k}{2k\atopwithdelims ()k}^3\equiv 0\pmod {p^{2r}}, \end{equation*}$$which was conjectured by Z.‐W. Sun in 2011, thus confirming Sun's conjecture. Further, applying a very‐well‐poised 6ϕ5$_6\phi _5$ summation and the creative microscoping method introduced by the author and Zudilin, we extend this supercongruence to the modulo p2r+1$p^{2r+1}$ case. We also give some similar results for primes p≡3(mod4)$p\equiv 3\pmod {4}$. Finally, we propose two conjectures on relevant supercongruences for further study. [ABSTRACT FROM AUTHOR]