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Concordant numbers within arithmetic progressions and elliptic curves

Author: Bo-Hae Im
Journal: Proc. Amer. Math. Soc. 141 (2013), 791-800
MSC (2010): Primary 11G05; Secondary 11D09, 11D45
Published electronically: July 13, 2012
MathSciNet review: 3003673
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Abstract: If the system of two diophantine equations $ X^2+mY^2=Z^2$ and $ X^2+nY^2=W^2$ has infinitely many integer solutions $ (X,Y,Z,W)$ with $ \operatorname {gcd}(X,Y)=1$, equivalently, the elliptic curve $ E_{m,n} : y^2=x(x+m)(x+n)$ has positive rank over $ \mathbb{Q}$, then $ (m,n)$ is called a strongly concordant pair. We prove that for a given positive integer $ M$ and an integer $ k$, the number of strongly concordant pairs $ (m, n)$ with $ m,n\in [1,N]$ and $ m,n \equiv k$ is at least $ O(N)$, and we give a parametrization of them.

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Additional Information

Bo-Hae Im
Affiliation: Department of Mathematics, Chung-Ang University, 221, Heukseok-dong, Dongjak-gu, Seoul, 156-756, South Korea

Received by editor(s): March 8, 2011
Received by editor(s) in revised form: July 19, 2011
Published electronically: July 13, 2012
Additional Notes: The author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (No. 2009-0087887).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.