On Some Congruence Conjectures Involving Binary Quadratic Forms

Abstract

In this paper, we mainly prove some congruence conjectures of Z.-H. Sun, for example, let $p$ be a prime such that $p\equiv 1(\mathrm{mod}6)$ and $p=x^2+3y^2$. Then $p{\sum }_{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{(3k+1)16^k}\equiv 4x^2-2p-\frac{p^2}{4x^2}(\mathrm{mod}{p}^3)$.