Some Complete Residue Systems in the Gaussian Integers - Tadee, Suton (Journal of Science and Science Education (JSSE), 2022)
In this paper, we study some complete residue systems in the Gaussian integers. By a complete residue system modulo where is a Gaussian integer, abbreviated by . The research results showed that for a positive integer , 1) <img title="{ x+yi : x,yZ,-n2≤ x<n2, -n2≤ y<n2 }" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n2≤&space;x<n2,&space;-n2≤&space;y<n2&space;&space;}"> is a , 2)<img title="{ x+yi : x,yZ,-n2< x≤ n2, -n2< y≤ n2 }" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n2<&space;x≤&space;n2,&space;-n2<&space;y≤&space;n2&space;&space;}"> is a , 3)<img title="{ x+yi : x,yZ,-n2< x≤ n2, -n2≤ y<n2 }" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n2<&space;x≤&space;n2,&space;-n2≤&space;y<n2&space;&space;}"> is a , 4)<img title="{ x+yi : x,yZ,-n2≤ x<n2, -n2< y≤ n2 }" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n2≤&space;x<n2,&space;-n2<&space;y≤&space;n2&space;&space;}"> is a , 5)<img title="{ x+yi : x,yZ,-n< x<n, | x | -n≤ y<-|x |+n}" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n<&space;x<n,&space;&space;|&space;x&space;&space;|&space;-n≤&space;y<-&space;|x&space;&space;|+n&space;}"> is a , 6)<img title="{ x+yi : x,yZ,-n< x<n, | x | -n< y≤ -|x |+n}" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n<&space;x<n,&space;&space;|&space;x&space;&space;|&space;-n<&space;y≤&space;-&space;|x&space;&space;|+n&space;}"> is a , 7)<img title="{ x+yi : x,yZ,-n< y<n, | y| -n< x≤ -|y |+n}" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n<&space;y<n,&space;&space;|&space;y&space;|&space;-n<&space;x≤&space;-&space;|y&space;&space;|+n&space;}"> is a , 8) <img title="{ x+yi : x,yZ,-n< y<n, | y| -n≤ x< -|y |+n}" src="https://latex.codecogs.com/gif.latex?&space;{&space;x+yi&space;:&space;x,y&space;Z,-n<&space;y<n,&space;&space;|&space;y&space;|&space;-n≤&space;x<&space;-&space;|y&space;&space;|+n&space;}"> is a .