The cubic congruence x(3)+ax(2)+bx+c equivalent to 0(mod p) and binary quadratic forms F(x, y) = ax(2)+bxy+cy(2)

Abstract

Let F(x,y) ax(2) + bxy + cy(2) be a binary quadratic form of discriminant Delta = b(2) - 4ac for a, b, c is an element of Z, let p be a prime number and let F-p be a finite field. In this paper we formulate the number of integer solutions of cubic congruence x(3) + ax(2) + bx + c equivalent to 0 (mod p) over Fp for two specific binary quadratic forms F-1(k) (x, y) = x(2) + kxy + ky(2) and F-2(k) (x, y) = kx(2) + kxy + k(2) y(2) for integer k such that 1 <= k <= 9. Later we consider representation of primes by F-1(k) and F-2(k).