Proof of some supercongruences concerning truncated hypergeometric series

Abstract

In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$\sum _{k=0}^{p^r-1}(3k+1)\frac{(\frac{1}{2}{)}_k^3}{(1{)}_k^3}{4}^k\equiv p^r+\frac{7}{6}{p}^{r+3}{B}_{p-3}(\mathrm{mod}{p}^{r+4})$$ and $$\sum _{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac{1}{2}{)}_k^4}{(1{)}_k^4}\equiv p^r+\frac{7}{6}{p}^{r+3}{B}_{p-3}(\mathrm{mod}{p}^{r+4}),$$ where $(x{)}_k=x(x+1)\cdots (x+k-1)$ is the Pochhammer symbol and $B_0,B_1,B_2,\dots$ are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. Appl. Math. 120 (2020), Art. 102078], respectively.

18 pages. This is a preliminary version