The Cubic Congreunce $x^3+a{x}^2+bx+c\equiv 0(\mathrm{mod}p)$ and Binary Quadratic Forms $F(x,y)=a{x}^2+bxy+c{y}^2$

Abstract

Let $F(x,y)=a{x}^2+bxy+c{y}^2$ be a binary quadratic form of discriminant $\mathrm{\Delta }=b^2–4ac$ for $a,b,c\in \mathbb{Z}$, let $p$ be a prime number and let ${\mathbb{F}}_p$ be a finite field. In this paper we formulate the number of integer solutions of cubic congruence $x^3+a{x}^2+bx+c\equiv 0(\mathrm{mod}p)$ over ${\mathbb{F}}_p$, for two specific binary quadratic forms $F_1^k(x,y)=x^2+kxy+k{y}^2$ and $F_2^k(x,y)=k{x}^2+kxy+k^2{y}^2$ for integer $k$ such that $1\le k\le 9$. Later we consider representation of primes by $F_1^k$ and $F_2^k$.