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Article: Diophantine inequalities with mixed powers

TitleDiophantine inequalities with mixed powers
Authors
Issue Date1982
PublisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jnt
Citation
Journal Of Number Theory, 1982, v. 15 n. 2, p. 149-163 How to Cite?
AbstractLet {λi}i = 1 s (s ≥ 2) be a finite sequence of non-zero real numbers, not all of the same sign and in which not all the ratios λi λj are rational. A given sequence of positive integers {ni}i = 1 s is said to have property (P) ((P*) respectively) if for any {λi}i = 1 s and any real number η, there exists a positive constant σ, depending on {λi}i = 1 s and {ni}i = 1 s only, so that the inequality |η + Σi = 1 s λixi ni| < (max xi)-σ has infinitely many solutions in positive integers (primes respectively) x1, x2,..., xs. In this paper, we prove the following result: Given a sequence of positive integers {ni}i = 1 ∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on {ni}i = j ∞ only, such that {ni}i = j j + s - 1 has property (P) (or (P*)), is that Σi = 1 ∞ ni -1 = ∞. These are parallel to some striking results of G. A. Freǐman, E. J. Scourfield and K. Thanigasalam. © 1982.
Persistent Identifierhttp://hdl.handle.net/10722/156228
ISSN
2015 Impact Factor: 0.596
2015 SCImago Journal Rankings: 0.858
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorTsang, KMen_US
dc.date.accessioned2012-08-08T08:40:55Z-
dc.date.available2012-08-08T08:40:55Z-
dc.date.issued1982en_US
dc.identifier.citationJournal Of Number Theory, 1982, v. 15 n. 2, p. 149-163en_US
dc.identifier.issn0022-314Xen_US
dc.identifier.urihttp://hdl.handle.net/10722/156228-
dc.description.abstractLet {λi}i = 1 s (s ≥ 2) be a finite sequence of non-zero real numbers, not all of the same sign and in which not all the ratios λi λj are rational. A given sequence of positive integers {ni}i = 1 s is said to have property (P) ((P*) respectively) if for any {λi}i = 1 s and any real number η, there exists a positive constant σ, depending on {λi}i = 1 s and {ni}i = 1 s only, so that the inequality |η + Σi = 1 s λixi ni| < (max xi)-σ has infinitely many solutions in positive integers (primes respectively) x1, x2,..., xs. In this paper, we prove the following result: Given a sequence of positive integers {ni}i = 1 ∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on {ni}i = j ∞ only, such that {ni}i = j j + s - 1 has property (P) (or (P*)), is that Σi = 1 ∞ ni -1 = ∞. These are parallel to some striking results of G. A. Freǐman, E. J. Scourfield and K. Thanigasalam. © 1982.en_US
dc.languageengen_US
dc.publisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jnten_US
dc.relation.ispartofJournal of Number Theoryen_US
dc.titleDiophantine inequalities with mixed powersen_US
dc.typeArticleen_US
dc.identifier.emailTsang, KM:kmtsang@maths.hku.hken_US
dc.identifier.authorityTsang, KM=rp00793en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/0022-314X(82)90022-1-
dc.identifier.scopuseid_2-s2.0-49049141648en_US
dc.identifier.volume15en_US
dc.identifier.issue2en_US
dc.identifier.spage149en_US
dc.identifier.epage163en_US
dc.identifier.isiWOS:A1982PN98100002-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridTsang, KM=7201554731en_US

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