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

Abstract

Let \(p\) be an odd prime. We consider the solution sets
\[
S_{+}(p^2)=\{n\in Z_{p^2}^{*} \mid n \equiv a+b \equiv a b\pmod {p^2}\}
\]
and
\[
S_{-}(p^2)=\{n \in Z_{p^2}^{*}\mid n \equiv a-b \equiv a b\pmod {p^2}\},
\]
where \(Z_{p^2}^{*}\) denote a reduced residue system modulo \({p^2}\). We also establish congruences about sum and product of the residues or quadratic residues in \(S_+(p^2)\) or in \(S_-(p^2)\) modulo \({p^2}\). Finally, we obtain the number of solution sets based on the classification of prime numbers, where \(a\) and \(b\) are quadratic residues or quadratic non-residues, respectively.