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=x2+2y2 equivalent to 1(mod8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=x<^>2+2y<^>2\equiv 1\ ({\textrm{mod}}\ 8)$$\end{document}. Then, p n-ary sumation k=0p-12kk2(8k+1)16k equivalent to 3p n-ary sumation k=0p-12kk2(8k+3)16k equivalent to 4x2-2p-p24x2(modp3).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p\sum _{k=0}<^>{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) <^>2}{(8k+1)16<^>k}\equiv 3p\sum _{k=0}<^>{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) <^>2}{(8k+3)16<^>k}\equiv 4x<^>2-2p-\frac{p<^>2}{4x<^>2}\ ({\textrm{mod}}\ p<^>3). \end{aligned}$$\end{document}