Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2021; 36(3): 401-411

Online first article February 17, 2021      Printed July 31, 2021

https://doi.org/10.4134/CKMS.c200247

Copyright © The Korean Mathematical Society.

Diophantine triple with Fibonacci numbers and elliptic curve

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).