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

2021/09/27 by Yulu Feng, Feng, Yulu, Shaofang Hong +1
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.12782

openalex publication_date 2021/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n≥ 1, e≥ 1, k≥ 2 and c be integers. An integer u is called a unit in the ring ℤ_n of residue classes modulo n if gcd(u, n)=1. A unit u is called an exceptional unit in the ring ℤ_n if gcd(1-u,n)=1. We denote by N_k,c,e(n) the number of solutions (x₁,...,x_k) of the congruence x₁^e+...+x_k^e≡ c \pmod n with all x_i being exceptional units in the ring ℤ_n. In 2017, Mollahajiaghaei presented a formula for the number of solutions (x₁,...,x_k) of the congruence x₁²+...+x_k²≡ c\pmod n with all x_i being the units in the ring ℤ_n. Meanwhile, Yang and Zhao gave an exact formula for 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 N_k,c,2(n). Our result extends Mollahajiaghaei's theorem and that of Yang and Zhao.

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