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Diophantine triple with Fibonacci numbers and elliptic curve. (English) Zbl 1486.11025

A Diophantine \(m\)-tuple is a set \(\{a_1,a_2,\dots, a_m\}\) of positive integers such that \(a_ia_j+1\) is a square for all \(1\leq i<j\leq m\). Given a Diophantine triple \(\{a,b,c\}\) it is natural to ask for which \(x\) it is possible to extend the Diophantine triple to a Diophantine quadruple \(\{a,b,c,x\}\). This happens only if \(x\) is the abscissa of an integral point \(P\) on the elliptic curve \(y^2=(ax+1)(bx+1)(cx+1)\).
In this paper, the author consider the Diophantine triple \(\{F_{2k},5F_{2k+2},3F_{2k}+7F_{2k+2}\}\), where \(F_k\) is the \(k\)-th Fibonacci number, and study the rational elliptic curve \(y^2=(F_{2k}x+1)(5F_{2k+2}x+1)((3F_{2k}+7F_{2k+2})x+1)\). Firstly, they study the rational torsion points on the curve. Then, they classify all the integral points on the elliptic curve, under the assumptions that \(k\) is an even integer with \(k\not\equiv 4\pmod 6\) and that the elliptic curve has rank \(1\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11G05 Elliptic curves over global fields
11D09 Quadratic and bilinear Diophantine equations

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[19] SIMATH manual, Saarbr¨ucken, 1997 Jinseo Park Department of Mathematics Education Catholic Kwandong University Gangneung 25601, Korea Email address:jspark@cku.ac
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