Sequences of consecutive squares on quartic elliptic curves

TL;DR

This paper constructs infinitely many quartic elliptic curves over rationals that contain sequences of at least six rational points with x-coordinates as consecutive squares, demonstrating the abundance and independence of such sequences.

Contribution

It introduces a method to generate infinitely many elliptic curves with long sequences of rational points having consecutive square x-coordinates, extending previous results.

Findings

Existence of infinitely many elliptic curves with 6-term consecutive square sequences.

The 6 rational points in these sequences are proven to be independent.

For any fixed 6-term sequence of consecutive squares, infinitely many elliptic curves realize it as rational points.

Abstract

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 $C(\mathbb Q)$.