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

Let p be an odd prime. We consider the solution sets S_+ (p^2) = \{ n ∈ Zp^2^* n ≡ a + b ≡ ab {p^2} \} and S_- (p^2) = \{ n ∈ Zp^2^* n ≡ a - b ≡ ab {p^2} \}, where Zp^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.