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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].
Mariusz Ska/lba. "On sets which contain a qth power residue for almost all prime modules." Colloquium Mathematicae 102.1 (2005): 67-71. <http://eudml.org/doc/283742>.
@article{MariuszSka2005, abstract = {A classical theorem of M. Fried [2] asserts that if non-zero integers $β₁,...,β_l$ have the property that for each prime number p there exists a quadratic residue $β_j$ 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 [3].}, author = {Mariusz Ska/lba}, journal = {Colloquium Mathematicae}, language = {eng}, number = {1}, pages = {67-71}, title = {On sets which contain a qth power residue for almost all prime modules}, url = {http://eudml.org/doc/283742}, volume = {102}, year = {2005}, }
TY - JOUR AU - Mariusz Ska/lba TI - On sets which contain a qth power residue for almost all prime modules JO - Colloquium Mathematicae PY - 2005 VL - 102 IS - 1 SP - 67 EP - 71 AB - A classical theorem of M. Fried [2] asserts that if non-zero integers $β₁,...,β_l$ have the property that for each prime number p there exists a quadratic residue $β_j$ 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 [3]. LA - eng UR - http://eudml.org/doc/283742 ER -