Diophantine m-tuples and elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, p. 111-124

A Diophantine m-tuple is a set of m positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form y 2 =(ax+1)(bx+1)(cx+1), where {a,b,c}, is a Diophantine triple. In particular, we consider the elliptic curve E k defined by the equation y 2 =(F 2k x+1)(F 2k+2 x+1)(F 2k+4 x+1), where k2 and F n , denotes the n-th Fibonacci number. We prove that if the rank of E k (𝐐) is equal to one, or k50, then all integer points on E k are given by(x,y){(0±1),(4F 2k+1 F 2k+2 F 2k+3 ±2F 2k+1 F 2k+2 -1×2F 2k+2 2 +12F 2k+2 F 2k+3 +1}.

Un m-uplet diophantien est un ensemble de m entiers naturels non nuls tel que le produit quelconque de deux d’entre eux augmenté de 1 est un carré parfait. Dans cet article, nous nous intéressons a certaines propriétés de courbes elliptiques d’équation du type y 2 =(ax+1)(bx+1)(cx+1), où {a,b,c} est un triplet diophantien. Nous considérons en particulier la courbe elliptique E k définie par l’équation y 2 =(F 2k x+1)(F 2k+2 x+1)(F 2k+4 x+1),k2 et F n désigne le n-ème nombre de Fibonacci. Nous montrons que si le rang de E k est égal a 1, ou si k50, alors les points entiers sur E k sont donnés par (x,y){(0±1),(4F 2k+1 F 2k+2 F 2k+3 ±2F 2k+1 F 2k+2 -1×2F 2k+2 2 +12F 2k+2 F 2k+3 +1}.

@article{JTNB_2001__13_1_111_0,
     author = {Dujella, Andrej},
     title = {Diophantine $m$-tuples and elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {111-124},
     zbl = {1046.11034},
     mrnumber = {1838074},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_111_0}
}
Dujella, Andrej. Diophantine $m$-tuples and elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 111-124. http://www.numdam.org/item/JTNB_2001__13_1_111_0/

[1] J. Arkin, V.E. Hoggatt, E.G. Strauss, On Euler's solution of a problem of Diophantus. Fibonacci Quart. 17 (1979), 333-339. | MR 550175 | Zbl 0418.10021

[2] A. Baker, H. Davenport, The equations 3x2 - 2 = y2 and 8x2 - 7 = z2. Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137. | MR 248079 | Zbl 0177.06802

[3] A. Baker, G. Wüstholz, Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 19-62. | MR 1234835 | Zbl 0788.11026

[4] J.H.E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations. Proc. Glasgow Math. Assoc. 7 (1965), 24-28. | MR 177944 | Zbl 0127.01902

[5] J.E. Cremona, Algorithms for Modular Elliptic Curves. Cambridge Univ. Press, 1997. | MR 1628193 | Zbl 0872.14041

[6] L.E. Dickson, History of the Theory of Numbers. Vol. 2, Chelsea, New York, 1966, pp. 513-520. | Zbl 0958.11500

[7] Diophantus Of Alexandria, Arithmetics and the Book of Polygonal Numbers. (I.G. Bashmakova, Ed.), Nauka, Moscow, 1974 (in Russian), pp. 103-104, 232.

[8] A. Dujella, On Diophantine quintuples. Acta Arith. 81 (1997), 69-79. | MR 1454157 | Zbl 0871.11019

[9] A. Dujella, The problem of the extension of a parametric family of Diophantine triples. Publ. Math. Debrecen 51 (1997), 311-322. | MR 1485226 | Zbl 0903.11010

[10] A. Dujella, A proof of the Hoggatt-Bergum conjecture. Proc. Amer. Math. Soc. 127 (1999), 1999-2005. | MR 1605956 | Zbl 0937.11011

[11] A. Dujella, A parametric family of elliptic curves. Acta Arith. 94 (2000), 87-101. | MR 1762457 | Zbl 0972.11048

[12] A. Dujella, Absolute bound for the size of Diophantine m-tuples. J. Number Theory, to appear. | MR 1838708 | Zbl 1010.11019

[13] A. Dujella, A. Pethö, A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306. | MR 1645552 | Zbl 0911.11018

[14] A. Dujella, A. Pethö, Integer points on a family of elliptic curves. Publ. Math. Debrecen 56 (2000), 321-335. | MR 1765985 | Zbl 0960.11019

[15] E. Herrmann, A. Pethö, H.G. Zimmer, On Fermat's quadruple equations. Abh. Math. Sem. Univ. Hamburg 69 (1999), 283-291. | MR 1722939 | Zbl 0952.11033

[16] V.E. Hoggatt, G.E. Bergum, A problem of Fermat and the Fibonacci sequence. Fibonacci Quart. 15(1977), 323-330. | MR 457339 | Zbl 0383.10007

[17] D. Husemöller, Elliptic Curves. Springer-Verlag, New York, 1987. | MR 868861 | Zbl 0605.14032

[18] B.W. Jones, A second variation on a problem of Diophantus and Davenport. Fibonacci Quart. 16 (1978), 155-165. | MR 498978 | Zbl 0382.10011

[19] K.S. Kedlaya, Solving constrained Pell equations. Math. Comp. 67 (1998), 833-842. | MR 1443123 | Zbl 0945.11027

[20] A. Knapp, Elliptic Curves. Princeton Univ. Press, 1992. | MR 1193029 | Zbl 0804.14013

[21] B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129-162. | MR 482230 | Zbl 0386.14009

[22] T. Nagell, Introduction to Number Theory. Almqvist, Stockholm; Wiley, New York, 1951. | MR 43111 | Zbl 0042.26702

[23] T. Nagell, Contributions to the theory of a category of Diophantine equations of the second degree with two unknowns. Nova Acta Soc. Sci. Upsal. 16 (1954), 1-38. | MR 70645 | Zbl 0057.28304

[24] K. Ono, Euler's concordant forms. Acta Arith. 78 (1996), 101-123. | MR 1424534 | Zbl 0863.11038

[25] A. Pethö, E. Herrmann, H.G. Zimmer, S-integral points on elliptic curves and Fermat's triple equations. In: Algorithmic Number Theory, (J. P. Buhler, ed.), Lecture Notes in Comput. Sci. 1423 (1998), 528-540. | Zbl 0920.11086

[26] SIMATH manual, Universität des Saarlandes, Saarbrücken, 1997.

[27] M. Vellupillai, The equations z2 - 3y2 = -2 and z2 - 6x2 = -5, in: A Collection of Manuscripts Related to the Fibonacci Sequence. (V. E. Hoggatt, M. Bicknell-Johnson, eds.), The Fibonacci Association, Santa Clara, 1980, pp. 71-75. | MR 624070 | Zbl 0511.00007

[28] D. Zagier, Elliptische Kurven: Fortschritte und Anwendungen. Jahresber. Deutsch. Math.-Verein 92 (1990), 58-76. | MR 1056202 | Zbl 0708.14019