On Binary Quadratic Forms Modulo n

Abstract

Given a binary quadratic polynomial $f(x_1,x_2)=\alpha x_1^2+\beta x_1x_2+\gamma x_2^2\in \mathbb {Z}[x_1,x_2]$, for every $c\in \mathbb Z$ and $n\ge 2$, we study the number of solutions $\mathrm {N}_J(f;c,n)$ of the congruence equation $f(x_1,x_2)\equiv c\bmod {n}$ in $(\mathbb {Z}/n\mathbb {Z})^2$ such that $x_i\in (\mathbb {Z}/n\mathbb {Z})^\times $ for $i\in J\subseteq \{1,2\}$.

Keywords

Binary quadratic form / Counting solutions / Congruence equation modulo n