Document Detail


Congruent numbers with many prime factors.
MedLine Citation:
PMID:  23213259     Owner:  NLM     Status:  PubMed-not-MEDLINE    
Abstract/OtherAbstract:
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].
Authors:
Ye Tian
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Publication Detail:
Type:  Journal Article; Research Support, Non-U.S. Gov't     Date:  2012-12-04
Journal Detail:
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:
Created Date:  2012-12-27     Completed Date:  2013-02-21     Revised Date:  2013-06-26    
Medline Journal Info:
Nlm Unique ID:  7505876     Medline TA:  Proc Natl Acad Sci U S A     Country:  United States    
Other Details:
Languages:  eng     Pagination:  21256-8     Citation Subset:  -    
Affiliation:
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:
Proc Natl Acad Sci U S A. 2012 Dec 26;109(52):21182-3   [PMID:  23248279 ]

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