On the addition of squares of units and nonunits modulo n

TextLet ZnZn be the ring of residue classes modulo n   and Zn⁎ be the group of its units. In 1926, Brauer obtained an explicit formula for the number of solutions of the linear congruence x1+⋯+xk≡c(modn) with x1,x2,…,xk∈Zn⁎. In 2009, for any c∈Znc∈Zn, Sander gave a formula for the number of representations of c as the sum of two units, the sum of two nonunits, and the sum of two mixed pairs, respectively. In this paper, we extend Sander's results to the quadratic case. We also pose some problems for further research.VideoFor a video summary of this paper, please visit http://youtu.be/PjC2lhc6Cs0.