整数的模n同因分类

The Mod N Equi-divisor Classification of Integers

在整数集Z上定义了模n同因关系,得到整数的模n同因分类Z(n).证明了:Z(n)的元素个数是T(n)(其中T(n)是n的正因数个数);Z(n)关于乘法[a][b]=[ab]作成以[0]为零元,[1]为单位元的交换半群,且除[1]外其余的元都没有逆元;在不等式T(n)+φ(n)≤n+1中,当且仅当n=1,4,p(p为素数)时等号成立,其中φ(n)是欧拉函数.

An equivalent divisor relation based on the modulus of integer n （mod n） is defined in the integer set Z. Amod n equi-divisor classification Z（n） of the integer n is therefore obtained. It is proved that the number of elements in Z（n） is T （n）, where T（n） is the number of positive divisors of n. Based on the multiplication [a] [b] = [ab], Z（n） forms a commutative semi-group, in which [0] is the zero element, [1] is the unit element and the only element having its inverse element. For the inequalityT （n）＋φ （n） ≤n＋1, the equality holds true if and Only if n=1, 4p（p is a prime number and φ（n） is an Euler function）.