Concordant numbers within arithmetic progressions and elliptic curves
HTML articles powered by AMS MathViewer
- by Bo-Hae Im PDF
- Proc. Amer. Math. Soc. 141 (2013), 791-800 Request permission
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.References
- Michael A. Bennett, Lucas’ square pyramid problem revisited, Acta Arith. 105 (2002), no. 4, 341–347. MR 1932567, DOI 10.4064/aa105-4-3
- Hubert Delange, On some sets of pairs of positive integers, J. Number Theory 1 (1969), 261–279. MR 242762, DOI 10.1016/0022-314X(69)90045-6
- Dale Husemoller, Elliptic curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987. With an appendix by Ruth Lawrence. MR 868861, DOI 10.1007/978-1-4757-5119-2
- B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
- Ken Ono, Euler’s concordant forms, Acta Arith. 78 (1996), no. 2, 101–123. MR 1424534, DOI 10.4064/aa-78-2-101-123
- Takashi Ono, Variations on a theme of Euler, The University Series in Mathematics, Plenum Press, New York, 1994. Quadratic forms, elliptic curves, and Hopf maps; Translated and revised from the second Japanese edition by the author; Appendix 2 by Masanari Kida. MR 1306948, DOI 10.1007/978-1-4757-2326-7
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- J. B. Tunnell, A classical Diophantine problem and modular forms of weight $3/2$, Invent. Math. 72 (1983), no. 2, 323–334. MR 700775, DOI 10.1007/BF01389327
Additional Information
- Bo-Hae Im
- Affiliation: Department of Mathematics, Chung-Ang University, 221, Heukseok-dong, Dongjak-gu, Seoul, 156-756, South Korea
- MR Author ID: 768467
- Email: bohaeim@gmail.com, imbh@cau.ac.kr
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 791-800
- MSC (2010): Primary 11G05; Secondary 11D09, 11D45
- DOI: https://doi.org/10.1090/S0002-9939-2012-11372-3
- MathSciNet review: 3003673