On sets which contain a $q$th power residue for almost all prime modules
Volume 102 / 2005
Colloquium Mathematicum 102 (2005), 67-71
MSC: 11R20, 11A15.
DOI: 10.4064/cm102-1-6
Abstract
A classical theorem of M. Fried \cite{fri} asserts that if non-zero integers $\beta_1,\ldots,\beta_l$ have the property that for each prime number $p$ there exists a quadratic residue $\beta_j$ 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 \cite{schiska}.
