On the solution set of additive and multiplicative congruences modulo primes

Abstract

Let \(p\) be an odd prime. In this paper, analogs of Wilson's and Wolstenholme's theorems on the solution sets
\[
S_{+}=\{n\in Z_{p}^{*} \mid n \equiv a+b \equiv a b\pmod p\}
\]
and
\[
S_{-}=\{n \in Z_{p}^{*}\mid n \equiv a-b \equiv a b\pmod p\}
\]
are given, where \(Z_{p}^{*}\) denote a reduced residue system modulo \(p\). We also establish congruences about sum and product of the quadratic residues in \(S_+\) or in \(S_-\) modulo \(p\). Finally, we raise a problem on how to solve Hadamard's conjecture in the last section.