| Congruent numbers with many prime factors. | |
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MedLine Citation:
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PMID: 23213259 Owner: NLM Status: PubMed-not-MEDLINE |
Abstract/OtherAbstract:
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Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite order on the elliptic curve my2 = x3 - x. Heegner constructed such rational points in the case that m are primes congruent to 5,7 modulo 8 or twice primes congruent to 3 modulo 8 [Monsky P (1990) Math Z 204:45-68]. We extend Heegner's result to integers m with many prime divisors and give a sketch in this report. The full details of all the proofs will be given in ref. 1 [Tian Y (2012) Congruent Numbers and Heegner Points, arXiv:1210.8231]. |
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Authors:
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Ye Tian |
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Publication Detail:
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Type: Journal Article; Research Support, Non-U.S. Gov't Date: 2012-12-04 |
Journal Detail:
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Title: Proceedings of the National Academy of Sciences of the United States of America Volume: 109 ISSN: 1091-6490 ISO Abbreviation: Proc. Natl. Acad. Sci. U.S.A. Publication Date: 2012 Dec |
Date Detail:
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Created Date: 2012-12-27 Completed Date: 2013-02-21 Revised Date: 2013-04-16 |
Medline Journal Info:
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Nlm Unique ID: 7505876 Medline TA: Proc Natl Acad Sci U S A Country: United States |
Other Details:
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Languages: eng Pagination: 21256-8 Citation Subset: - |
Affiliation:
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Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. ytian@math.ac.cn |
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Comment In:
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Proc Natl Acad Sci U S A. 2012 Dec 26;109(52):21182-3
[PMID:
23248279
]
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From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
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