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

Abstract: In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv3\pmod 4$ be a prime. Then $$\sum_{k=0}^{{p-1}\frac{\binom{2k}k2}{8k}\equiv-\sum_{k=0}{p-1}\frac{\binom{2k}k2}{(-16)k}\pmod{p3}.$$} And for any odd prime $p$, if $p=x^2+y2$ with $4|x-1, 2|y$, then $$ \sum_{k=0}^{{p-1}\frac{(k+1)\binom{2k}k2}{8k}+\sum_{k=0}{(p-1)/2}\frac{(2k+1)\binom{2k}k2}{(-16)k}\equiv2\left(\frac{2}p\right)x\pmod{p3}.} $$