OBTAINING ALL OR HALF OF Un AS !x' × !x + 1'

摘要

For any odd positive integer n, let Un denote the set of units of Zn. We investigate for which values of n the set Un is given by the direct product !x" × !x + 1" for one or more values x, and also which are such that !x" × !x + 1" likewise contains precisely half of the elements of Un. We give theorems for odd values of n that are (a) prime or prime power, (b) of the form pi qj (i ≥ 1, j ≥ 1), and (c) of the form 3pq. We provide tables giving details for all odd values of n in the range 2 < n < 300.