本项目要研究和解决孙智伟和申请人提出的许多涉及素数的二次型表示的超同余式猜想，特别要确定一批包含二项式系数之和式及包含Apery多项式之和式模奇素数平方之同余式.我们的方法工具是Jacobi多项式，Legendre多项式，超几何级数变换，Jacobsthal和，Zeilberger算法及有限域椭圆曲线理论。通过综合利用数论、组合、分析方法，特别是利用Morton和Ishii关于有限域上椭圆曲线点数的工作以及各种超几何级数变换公式，我们将揭示Legendre多项式及更一般的Jacobi多项式与类数为1,2或4的二次型及超同余式的联系,通过同余式构造素数二次型表示中的参数。我们也研究与二次型无关的超同余式，力图解决孙智伟和Zudilin提出的相关猜想。

The purpose of the project is to study and solve many conjectures on congruences concerning binary quadratic forms posed by Zhi-Wei Sun and the applicant. In particular, we will determine certain sums involving binomial coefficients or Apery polynomials modulo the square of a given odd prime. Our tools are Jacobi polynomials, Legendre polynomials, hypergeometric series transformations, Jacobsthal sums, Zeilberger's algorithm and the theory of elliptic curves over finite fields. By combining methods from number theory, combinatorics and analysis, and using the work of Morton and Ishii on the number of points on elliptic curves over finite fields and various transformation formulas for hypergeometric series, we will reveal the connections among Legendre polynomials (Jacobi polynomials), binary quadratic forms with class number 1,2 or 4 and super congruences, and construct the parameters in the representations of primes by certain binary quadratic forms via congruences. In the project, we also investigate some super congruences not involving quadratic forms and solve some related conjectures posed by Zhi-Wei Sun and Zudilin.