On the solution set of additive and multiplicative congruences modulo primes II

Abstract

Let \(p>3\) be a prime and
\[
S_{+}=\{n\in Z_{p}^{*} \mid n \equiv a+b \equiv a b\pmod p\}
\]
and
\[
S_{-}=\{n\in Z_{p}^{*} \mid n \equiv a-b \equiv a b\pmod p\},
\]
where \(Z_{p}^{*}\) denote a reduced residue system modulo \({p}\). In this work, we investigate the solution sets \(\left|S_{+}\cap S_{-}\right|\), \( \left|\{n|\pm n\in S_{+}\}\right|\) and \( \left|\{n|n\in S_{+}~ \text{and}~ n+4k\in S_{+}\}\right|\) for some given integer \(k\). Meanwhile, we consider how many \(n\in Z_{p}^{*}\) such that \(n\) can be written in both linear and quadratic forms of two variables. And how many \(n\in Z_{p}^{*}\) can be written as different quadratic forms of two variables.