Connections between binomial coefficients and binary quadratic forms

In this paper, we mainly prove some congruences involving binomial coefficients and binary quadratic forms. One such example is the following: Let p b be a prime such that \(p=x^2+2y^2\equiv 1\ ({\textrm{mod}}\ 8)\) $p=x^2+2y^2\equiv 1(\text{mod}8)$ . Then, \(\begin{aligned} p\sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2}{(8k+1)16^k}\equiv 3p\sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2}{(8k+3)16^k}\equiv 4x^2-2p-\frac{p^2}{4x^2}\ ({\textrm{mod}}\ p^3). \end{aligned}\) $$\begin{array}{c}\hfill p\sum _{k=0}^{p-1}\frac{{\left(\begin{array}{c}2k\\ k\end{array}\right)}^2}{(8k+1){16}^k}\equiv 3p\sum _{k=0}^{p-1}\frac{{\left(\begin{array}{c}2k\\ k\end{array}\right)}^2}{(8k+3){16}^k}\equiv 4x^2-2p-\frac{p^2}{4x^2}(\text{mod}{p}^3).\end{array}$$