Integer points on the elliptic curves induced by Diophantine triples
Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 745-757
https://doi.org/10.4134/CKMS.c190364
Published online July 5, 2020
Jinseo Park
Catholic Kwandong University
Abstract : A set $\{a_1, a_2, \dots, a_m\}$ of positive integers is called a Diophantine $m$-tuple if $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. In this paper, we find the structure of a torsion group of elliptic curves $E_k$ constructed by a Diophantine triple $\{F_{2k}, F_{2k+2}, 4F_{2k+1}F_{2k+2}F_{2k+3}\}$, and find all integer points on the elliptic curve under assumption that rank$(E_k(\mathbb{Q}))=2$.
Keywords : Diophantine $m$-tuple, Fibonacci numbers, elliptic curve
MSC numbers : Primary 11B39, 11G05, 11D09; Secondary 11D45
Supported by : This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1G1A1006396)
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