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 + 2 y 2 ≡ 1 (mod 8) . Then, p ∑ k = 0 p - 1 2 k k 2 (8 k + 1) 16 k ≡ 3 p ∑ k = 0 p - 1 2 k k 2 (8 k + 3) 16 k ≡ 4 x 2 - 2 p - p 2 4 x 2 (mod p 3). [ABSTRACT FROM AUTHOR]