On sets which contain a $q$th power residue for almost all prime modules

Mariusz Ska/lba¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21 00-956 Warszawa, Poland

Colloquium Mathematicum, Tome 102 (2005) no. 1, pp. 67-71

Résumé

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}.

Détail

DOI : 10.4064/cm102-1-6

Keywords: classical theorem fried cite fri asserts non zero integers beta ldots beta have property each prime number there exists quadratic residue beta mod certain product odd number square provide generalizations power residues degree cases prime power odd prime proofs involve combinatorial properties finite abelian groups arithmetic results cite schiska