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Irregular Diophantine \(m\)-tuples and elliptic curves of high rank. (English) Zbl 0964.11027

A set of \(m\) non-zero rational numbers \(\{a_1,\ldots,a_m\}\) is called a Diophantine \(m\)-tuple if for all \(1\leq i<j\leq m\) the rational number \(a_ia_j+1\) is a perfect square. In this very short note the author interprets certain properties of Diophantine \(4\)-tuples and \(5\)-tuples in terms of the existence of some relations connecting rational points of elliptic curves defined over \({\mathbb Q}\). As an application some elliptic curves over \({\mathbb Q}\) are found, whose Mordell-Weil group has rank \(8\) and torsion \(({\mathbb Z}/2{\mathbb Z})^2\).

MSC:

11G05 Elliptic curves over global fields
11B99 Sequences and sets
14H52 Elliptic curves

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[1] Arkin, J., Hoggatt, V. E., and Strauss, E. G.: On Euler’s solution of a problem of Diophantus. Fibonacci Quart., 17 , 333-339 (1979). · Zbl 0418.10021
[2] Cremona, J. E.: Algorithms for Modular Elliptic Curves. Cambridge Univ. Press, Cambridge-New York (1997). · Zbl 0872.14041
[3] Diophantus of Alexandria: Arithmetics and the Book of Polygonal Numbers (ed. Bashmakova, I. G.). Nauka, Moscow, pp. 103-104, 232 (1974) (in Russian).
[4] Dujella, A.: On Diophantine quintuples. Acta Arith., 81 , 69-79 (1997). · Zbl 0871.11019
[5] Dujella, A.: Diophantine triples and construction of high-rank elliptic curves over \(\bQ\) with three non-trivial 2-torsion points. Rocky Mountain J. Math. (to appear). · Zbl 0989.11032 · doi:10.1216/rmjm/1022008982
[6] Gibbs, P.: Some rational Diophantine sextuples. math.NT/9902081 (preprint). · Zbl 1131.11019 · doi:10.3336/gm.41.2.02
[7] Gibbs, P.: A generalised Stern-Brocot tree from regular Diophantine quadruples. math.NT/9903035 (preprint).
[8] Kihara, S.: On the rank of elliptic curves with three rational points of order 2. Proc. Japan Acad., 73A , 77-78 (1997). · Zbl 0906.11024 · doi:10.3792/pjaa.73.77
[9] Kihara, S.: On the rank of elliptic curves with three rational points of order 2. II. Proc. Japan Acad., 73A , 151 (1997). · Zbl 0906.11025
[10] Kulesz, L.: Courbes elliptiques de rang élevé, possédant un sous-groupe de torsion non trivial sur \(\bQ\) (preprint).
[11] Kulesz, L.: Courbes elliptiques de rang \(\geq 5\) sur \(\bQ(t)\) avec un groupe de torsion isomorphe á \(\bZ/2\bZ\times \bZ/2\bZ\). C. R. Acad. Sci. Paris Sér. I Math., 329 (6), 503-506 (1999). · Zbl 0954.11021 · doi:10.1016/S0764-4442(00)80050-6
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