Browse > Article
http://dx.doi.org/10.4134/CKMS.c200247

DIOPHANTINE TRIPLE WITH FIBONACCI NUMBERS AND ELLIPTIC CURVE  

Park, Jinseo (Department of Mathematics Education Catholic Kwandong University)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.3, 2021 , pp. 401-411 More about this Journal
Abstract
A Diophantine m-tuple is a set {a1, a2, …, am} of positive integers such that aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m. Let Ek be the elliptic curve induced by Diophantine triple {F2k, 5F2k+2, 3F2k + 7F2k+2}. In this paper, we find the structure of a torsion group of Ek, and find all integer points on Ek under assumption that rank(Ek(ℚ)) = 1 and some further conditions.
Keywords
Diophantine m-tuple; Fibonacci numbers; elliptic curve;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. E. Cremona, Algorithms for Modular Elliptic Curves, second edition, Cambridge University Press, Cambridge, 1997.
2 A. Dujella, Diophantine m-tuples and elliptic curves, J. Th'eor. Nombres Bordeaux 13 (2001), no. 1, 111-124.   DOI
3 A. Dujella and A. Petho, Integer points on a family of elliptic curves, Publ. Math. Debrecen 56 (2000), no. 3-4, 321-335.
4 D. Husemoller, Elliptic Curves, Graduate Texts in Mathematics, 111, Springer-Verlag, New York, 1987. https://doi.org/10.1007/978-1-4757-5119-2
5 A. Kim, Square Fibonacci numbers and square Lucas numbers, Asian Res. J. Math. 3 (2017), no. 3, 1-8.   DOI
6 V. E. Hoggatt, Jr., and G. E. Bergum, A problem of Fermat and the Fibonacci sequence, Fibonacci Quart. 15 (1977), no. 4, 323-330.
7 A. W. Knapp, Elliptic curves, Mathematical Notes, 40, Princeton University Press, Princeton, NJ, 1992.
8 A. Baker and H. Davenport, The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137. https://doi.org/10.1093/qmath/20.1.129   DOI
9 J. H. E. Cohn, Square Fibonacci Numbers, etc, Fibonacci Quart. 2 (1964), 109-113.
10 A. Dujella, A proof of the Hoggatt-Bergum conjecture, Proc. Amer. Math. Soc. 127 (1999), no. 7, 1999-2005. https://doi.org/10.1090/S0002-9939-99-04875-3   DOI
11 A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), no. 1, 87-101. https://doi.org/10.4064/aa-94-1-87-101   DOI
12 A. Dujella, Diophantine quadruples and Fibonacci numbers, Bull. Kerala Math. Assoc. 1 (2004), no. 2, 133-147.
13 A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 195, 291-306. https://doi.org/10.1093/qjmath/49.195.291   DOI
14 Y. Fujita, The Hoggatt-Bergum conjecture on D(-1)-triples {F2k+1, F2k+3, F2k+5} and integer points on the attached elliptic curves, Rocky Mountain J. Math. 39 (2009), no. 6, 1907-1932. https://doi.org/10.1216/RMJ-2009-39-6-1907   DOI
15 B. He, A. Togbe, and V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), no. 9, 6665-6709. https://doi.org/10.1090/tran/7573   DOI
16 J. Morgado, Generalization of a result of Hoggatt and Bergum on Fibonacci numbers, Portugal. Math. 42 (1983/84), no. 4, 441-445.
17 J. Park, Integer points on the elliptic curves induced by Diophantine triples, Commun. Korean Math. Soc. 35 (2020), no. 3, 745-757. https://doi.org/10.4134/CKMS.c190364   DOI
18 SIMATH manual, Saarbrucken, 1997
19 K. Ono, Euler's concordant forms, Acta Arith. 78 (1996), no. 2, 101-123. https://doi.org/10.4064/aa-78-2-101-123   DOI