Diophantine triple with Fibonacci numbers and elliptic curve
Commun. Korean Math. Soc. 2021 Vol. 36, No. 3, 401-411
https://doi.org/10.4134/CKMS.c200247
Published online February 17, 2021
Printed July 31, 2021
Jinseo Park
Catholic Kwandong University
Abstract : 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 perfect square for all $1\leq i < j \leq m$. Let $E_k$ be the elliptic curve induced by Diophantine triple $\{F_{2k}, 5F_{2k+2}, 3F_{2k}+7F_{2k+2}\}$. In this paper, we find the structure of a torsion group of $E_k$, and find all integer points on $E_k$ under assumption that rank$(E_k(\mathbb{Q}))=1$ and some further conditions.
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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