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].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-6,
author = {Mariusz Ska/lba},
title = {On sets which contain a qth power residue for almost all prime modules},
journal = {Colloquium Mathematicae},
volume = {103},
year = {2005},
pages = {67-71},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-6}
}
Mariusz Ska/lba. On sets which contain a qth power residue for almost all prime modules. Colloquium Mathematicae, Tome 103 (2005) pp. 67-71. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-6/