On the addition of two weighted squares of units mod n

Abstract

For any positive integer [Formula: see text], let [Formula: see text] be the ring of residue classes modulo [Formula: see text] and [Formula: see text] be the group of its units. Recently, for any [Formula: see text], Yang and Tang obtained a formula for the number of solutions of the quadratic congruence [Formula: see text] with [Formula: see text] units, nonunits and mixed pairs, respectively. In this paper, for any [Formula: see text], we give a formula for the number of representations of [Formula: see text] as the sum of two weighted squares of units modulo [Formula: see text]. We resolve a problem recently posed by Yang and Tang.