Proof of two congruence conjectures of Z.-W. Sun

Abstract

In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv 3(\mathrm{mod}4)$ be a prime. Then $$\sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{8^k}\equiv -\sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{(-16{)}^k}(\mathrm{mod}{p}^3).$$ And for any odd prime $p$, if $p=x^2+y^2$ with $4|x-1,2|y$, then $$\sum _{k=0}^{p-1}\frac{(k+1){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{8^k}+\sum _{k=0}^{(p-1)/2}\frac{(2k+1){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{(-16{)}^k}\equiv 2\left(\frac{2}{p}\right)x(\mathrm{mod}{p}^3).$$

28 pages, comments are welcome!