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Sequences of Consecutive Squares on Quartic Elliptic Curves by Sadek, Mohammad; Kamel, Mohamed is a scholarly article available to read on EtoBox.

What is Sequences of Consecutive Squares on Quartic Elliptic Curves about?

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 $C(\mathbb Q)$.

Author

Sadek, Mohammad; Kamel, Mohamed

Published

2018

Language

EN