On the addition of squares of units modulo n

Let ZnZn be the ring of residue classes modulo n  , and let Zn⁎ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of c∈Znc∈Zn as the sum of k units. Recently, Yang and Tang (2015) [6] gave a formula for the number of solutions of the equation x12+x22=c with x1,x2∈Zn⁎. In this paper, we generalize this result. We find an explicit formula for the number of solutions of the equation x12+⋯+xk2=c with x1,…,xk∈Zn⁎.