On the addition of values of a quadratic polynomial at units modulo n

Abstract

Let $$\mathbb{Z}_{n}$$ be the ring of residue classes modulo n, $$\mathbb{Z}_{n}^{*}$$ be its unit group, and let $$f(z)=az^{2}+bz$$ be an integral quadratic polynomial with $$b\neq0$$ and $${\rm gcd}(a,b)=1$$ . In this paper, for any integer c and positive integer n we give a formula for the number of solutions of the congruence equation $$f(x)+f(y)\equiv c({\rm mod}\, n)$$ with x, y units $$({\rm mod}\, n)$$ . This partly solves a problem posed by Yang and Tang in [11].