On sets which contain a qth power residue for almost all prime modules
A classical theorem of M. Fried [2] asserts that if non-zero integers have the property that for each prime number p there exists a quadratic residue mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].