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http://dx.doi.org/10.14403/jcms.2020.33.2.227

RELATION BETWEEN DIOPHANTINE TRIPLE AND ELLIPTIC CURVE  

Park, Jinseo (Department of Mathematics Education Catholic Kwandong University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.33, no.2, 2020 , pp. 227-236 More about this Journal
Abstract
A set {a1, a2,…, am} of positive integers is called Diophantine m-tuple if aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m. In this paper, we find the structure of torsion group of elliptic curve Ek constructed by Diophantine triple, and find all integer points on Ek under assumption that rank(Ek(ℚ)) = 1.
Keywords
Diophantine m-tuple; Elliptic curve; Torsion group; Rank;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, 1997.
2 L. E. Dickson, History of the Theory of Numbers Vol.2, Chelsea, New York, 1966.
3 A. Dujella, A parametric family of elliptic curves, Acta Arith., 94 (2000), no. 1, 87-101.   DOI
4 A. Dujella, Diophantine m-tuples and elliptic curves, 21st Journees Arithmetiques (Rome, 2001), J. Theor. Nombres Bordeaux, 13 (2001), no. 1, 111-124.   DOI
5 A. Dujella and A. Petho, Integer points on a family of elliptic curves, Publ. Math. Debrecen, 56 (2000), 321-335.
6 A. Dujella and M. Milic, On the torsion group of elliptic curves induced by D(4)-triples, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat., 22 (2014), 79-90.
7 A. Filipin, Y. Fujita, and A. Togbe, The extendibility of Diophantine pairs I: The general case, Glas. Mat. Ser. III, 49 (69) (2014), no. 1, 25-36.   DOI
8 Y. Fujita, The Hoggatt-Bergum conjecture on D(-1)-triples {$F_{2k+1}$, $F_{2k+3}$, $F_{2k+5}$} and integer points on the attached elliptic curves, Rocky Mountain J. Math., 39 (2009), no. 6, 1907-1932.   DOI
9 B. He, K. Pu, R. Shen, and A. Togbe, A note on the regularity of the Diophantine pair {k, 4k ${\pm}$ 4}, J. Theor. Nombres Bordeaux, 30 (2018), 879-892.   DOI
10 B. He, A. Togbe, and V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc., 371 (2019), 6665-6709.   DOI
11 D. Husemoller, Elliptic curves, Springer-Verlag, New York, 1987.
12 A. Knapp, Elliptic curves, Princeton Univ. Press, 1992.
13 J. B. Lee and J. Park, Some conditions on the form of third element from Diophantine pairs and its application, J. Korean Math. Soc., 55 (2018), no. 2, 425-445   DOI
14 M. Mikic, On the Mordell-Weil group of elliptic curves induced by the families of Diophantine triples, Rocky Mountain J. Math., 45 (2015), 1565-1589.   DOI
15 F. Najman, Integer points on two families of elliptic curves, Publ. Math. Debrecen, 75 (2009), 401-418.
16 F. Najman, Compact representation of quadratic integers and integer points on some elliptic curves, Rochy Mountain J. Math., 40 (2010), 1979-2002.
17 K. Ono, Euler's concordant forms, Acta Arith., 78 (1996), 101-123.   DOI
18 SIMATH manual, Saarbrucken, 1997