Elliptic curves and the density of theta-congruent numbers and concordant pairs in ratios

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dc.contributor.authorIm, Bo-Haeko
dc.date.accessioned2016-09-08T00:51:31Z-
dc.date.available2016-09-08T00:51:31Z-
dc.date.created2016-09-07-
dc.date.created2016-09-07-
dc.date.issued2014-01-
dc.identifier.citationJOURNAL OF PURE AND APPLIED ALGEBRA, v.218, no.1, pp.18 - 26-
dc.identifier.issn0022-4049-
dc.identifier.urihttp://hdl.handle.net/10203/212934-
dc.description.abstractIf k and l are coprime distinct integers, then we show that there exist infinitely many square-free integers n such that the system of two diophantine quadratic equations X-2 + knY(2) = Z(2), X-2 + lnY(2) = W-2 has infinitely many integer solutions (X, Y, Z, W) with gcd(X, Y) = 1, equivalently, the elliptic curve E-kn,E-ln : y(2) = x(x + kn)(x + ln) has positive rank over Q. (Such a pair (kn, ln) is called a strongly concordant pair.) Also, we give parametrizations of an infinite family of strongly concordant pairs (m, n) with ratio m/n = k/l and the corresponding integer solutions to X-2 + mY(2) = Z(2), X-2 + nY(2) = W-2. As an application, the result gives a parametrization of theta-congruent numbers as square-free parts of a parametrization S(t) and shows that if the number of t is an element of [1, N] such that S(t) itself is a theta-congruent number is not zero, then for all sufficiently large N, it is cN + O (N2/3+epsilon), where c > 0 and epsilon is an arbitrary small positive number. (C) 2013 Elsevier B.V. All rights reserved-
dc.languageEnglish-
dc.publisherELSEVIER SCIENCE BV-
dc.subjectFORMS-
dc.titleElliptic curves and the density of theta-congruent numbers and concordant pairs in ratios-
dc.typeArticle-
dc.identifier.wosid000327362900002-
dc.identifier.scopusid2-s2.0-84886639808-
dc.type.rimsART-
dc.citation.volume218-
dc.citation.issue1-
dc.citation.beginningpage18-
dc.citation.endingpage26-
dc.citation.publicationnameJOURNAL OF PURE AND APPLIED ALGEBRA-
dc.identifier.doi10.1016/j.jpaa.2013.04.007-
dc.contributor.localauthorIm, Bo-Hae-
dc.type.journalArticleArticle-
dc.subject.keywordPlusFORMS-
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