On the addition of units and nonunits mod m

Abstract

The group of units in the ring Z m of residue classes mod m consists of the residues a mod m with (a, m) = 1. We determine the number of representations of a fixed residue class mod m as the sum of two units in Z m , the sum of two nonunits, and the sum of mixed pairs, respectively.

On One Kind of Character Sums and Its Applications

“…where g is a primitive root mod p, d is uniquely determined by 4p � d 2 + 27b 2 with d ≡ 1 (mod 3), and k is any integer with 1 ≤ k ≤ p − 1/3. Many scholars have studied equations modulo a prime number and obtained a series of interesting results (see [16][17][18][19]).…”