On the solution set of additive and multiplicative congruences modulo primes

Zhongyan Shen


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\}
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.


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