Year: 2024
Type: article
Abstract: Let n be a natural number greater than 2 and q be the smallest prime dividing n. We show that a finite subset A of rationals, of cardinality at most q, contains a nth power in [Formula: see text] for almost every prime p if and only if A contains a perfect nth power, barring some exceptions when n is even. This generalizes the Grunwald–Wang theorem for nth powers, from one rational number to finite subsets of rational numbers. We also show that the upper bound q in this generalization is optimal for every n.
Author: Bhawesh Mishra
Institution: University of Memphis
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403002941