Let p be an odd prime. We consider the solution sets S_+ (p²) = \ n Zℂ^* n a + b ab p² \ and S_- (p²) = \ n Zℂ^* n a - b ab p² \, where Zℂ^* denote a reduced residue system modulo p². We also establish congruences about sum and product of the residues or quadratic residues in S_+ (p²) or in S_- (p²) modulo p². 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.
Shen et al. (2025) studied this question.