In a commutative ring R with unity, a unit u is called exceptional if \(u-1\) is also a unit. For \(R = {\mathbb {Z}}/n{\mathbb {Z}}\) and for any \(f(X) \in {\mathbb {Z}}[X]\), an element \({\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}\) is called an “f-exunit” if \(gcd(f(u),n) = 1\). Recently, we obtained the number of representations of a non-zero element of \({\mathbb {Z}}/n{\mathbb {Z}}\) as a sum of two f-exunits for a particular infinite family of polynomials \(f(X) \in {\mathbb {Z}}[X]\). In this paper, we complete this problem by proving a similar formula for any non-constant polynomial \(f(X) \in {\mathbb {Z}}[X]\).

在具有1的交换环R中，如果\（u-1 \）也是一个单位，则将u称为例外。对于\（R = {\ mathbb {Z}} / n {\ mathbb {Z}} \）和任何\（f（X）\ in {\ mathbb {Z}} [X] \）中，元素\如果\（gcd（f（u），n）= 1，则{{\ overline {u}} \ in {\ mathbb {Z}} / n {\ mathbb {Z}} \中的\）被称为“ f -exunit” \）。最近，对于一个特定的无限多项式族，我们获得了一个非零元素\（{\ mathbb {Z}} / n {\ mathbb {Z}} \）的表示形式，作为两个f单位之和。\（f（X）\ in {\ mathbb {Z}} [X] \）。在本文中，我们通过为{\ mathbb {Z}} [X] \）中的任何非常数多项式\（f（X）\）证明相似的公式来解决此问题。