On the sums of squares of exceptional units in residue class rings

Abstract

Let \(n\ge 1, e\ge 1, k\ge 2\) and c be integers. An integer u is called a unit in the ring \({\mathbb {Z}}_n\) of residue classes modulo n if \(\gcd (u, n)=1\). A unit u is called an exceptional unit in the ring \({\mathbb {Z}}_n\) if \(\gcd (1-u,n)=1\). We denote by \({\mathcal {N}}_{k,c,e}(n)\) the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^e+\cdots +x_k^e\equiv c \pmod n\) with all \(x_i\) being exceptional units in the ring \({\mathbb {Z}}_n\). In 2017, Mollahajiaghaei presented a formula for the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^2+\cdots +x_k^2\equiv c\pmod n\) with all \(x_i\) being the units in the ring \({\mathbb {Z}}_n\). Meanwhile, Yang and Zhao gave an exact formula for \({\mathcal {N}}_{k,c,1}(n)\). In this paper, by using Hensel’s lemma, exponential sums and quadratic Gauss sums, we derive an explicit formula for the number \({\mathcal {N}}_{k,c,2}(n)\). Our result extends Mollahajiaghaei’s theorem and that of Yang and Zhao.