Open Access
June 2019 Sequences of consecutive squares on quartic elliptic curves
Mohamed Kamel, Mohammad Sadek
Funct. Approx. Comment. Math. 60(2): 245-252 (June 2019). DOI: 10.7169/facm/1740


Let $C: y^2=ax^4+bx^2+c$, be an elliptic curve defined over $\mathbb{Q}$. A set of rational points $(x_i,y_i) \in C(\mathbb{Q})$, $i=1,2,\cdots,$ is said to be a sequence of consecutive squares if $x_i= (u + i)^2$, $i=1,2,\cdots$, for some $u\in \mathbb{Q}$. Using ideas of Mestre, we construct infinitely many elliptic curves $C$ with sequences of consecutive squares of length at least $6$. It turns out that these $6$ rational points are independent. We then strengthen this result by proving that for a fixed $6$-term sequence of consecutive squares, there are infinitely many elliptic curves $C$ with the latter sequence forming the $x$-coordinates of six rational points in $C(\mathbb{Q})$.


Download Citation

Mohamed Kamel. Mohammad Sadek. "Sequences of consecutive squares on quartic elliptic curves." Funct. Approx. Comment. Math. 60 (2) 245 - 252, June 2019.


Published: June 2019
First available in Project Euclid: 26 June 2018

zbMATH: 07068534
MathSciNet: MR3964263
Digital Object Identifier: 10.7169/facm/1740

Primary: 14G05
Secondary: 11B83

Keywords: Elliptic curves , rational points , sequences of consecutive squares

Rights: Copyright © 2019 Adam Mickiewicz University

Vol.60 • No. 2 • June 2019
Back to Top