Complete Residue Systems in the Gaussian Integers

Abstract

1. As a matter of notation small case Latin letters will represent real integers and Greek letters will represent Gaussian integers. There are many representations for the complete residue system modulo n but the one that usually comes to mind is the set of integers {0, 1, 2, *..., n-1 } . Other representations are sometimes used; for example if (a, n) = 1 then {a, 2a, 3a, *, na } is a representation of the complete residue system modulo n. Another representation which has a somewhat pleasing quality is found in Uspensky and Heaslet [1] and is { x -n/2 <x < n/2 } which is the representation with least absolute values. Consider the meaning of divisibility and congruences in the Gaussian integers. Recall that in the Gaussian integers 1 ,I means there is a Gaussian integer a such that a * y= =, and a = -3 (mod y) means that I a -of-. This congruence relation is an equivalence relation and, analogous to the real case, it is reasonable to define the complete residue system modulo y as the set of equivalence classes formed from the Gaussian integers with respect to the principal ideal ('y). The complete residue system modulo y will be abbreviated as CRS (mod y). Without any loss of generality y can be restricted to being in the first quadrant or on the positive real line. It is the purpose of this paper to exhibit several representations for the CRS (mod y). These representations are given in section 2 and the verification that they are representations of the complete residue systems is given in section 3.