SUMS OF POLYNOMIAL-TYPE EXCEPTIONAL UNITS MODULO

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. $n, k$ and $c$ integers such that $n\ge 1$ $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In this paper, we use principle cross-classification to derive explicit formula for number ${\mathcal N}_{k,f,c}(n)$ solutions $(x_1,...,x_k)$ congruence $x_1+...+x_k\equiv c\pmod n$ with all $x_i$ being $f$-exunits $\mathbb{Z}_n$. This extends recent result Anand {\it et al.} [On question $\mathbb{Z}/{n\mathbb{Z}}$, Arch. Math. (Basel)} {\bf 116} (2021), 403-409]. We more when $f(x)$ linear or quadratic.