On sequences of consecutive squares on elliptic curves

TL;DR

This paper constructs an infinite family of elliptic curves over rationals that contain a 5-term sequence of consecutive squares, demonstrating these sequences correspond to five independent rational points and establishing a lower bound on the rank.

Contribution

It introduces a method to generate elliptic curves with explicit 5-term sequences of consecutive squares and proves these sequences yield five independent rational points.

Findings

Constructed infinite family of elliptic curves with 5-term consecutive squares

Proved these sequences produce five independent rational points

Established that the rank of these curves is at least 5

Abstract

Let $C$ be an elliptic curve defined over $\mathbb Q$ by the equation $y^2=x^3+Ax+B$ where $A,B\in\mathbb Q$. A sequence of rational points $(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$ is said to form a sequence of consecutive squares on $C$ if the sequence of $x$-coordinates, $x_i,i=1,2,\ldots$, consists of consecutive squares. We produce an infinite family of elliptic curves $C$ with a $5$-term sequence of consecutive squares. Furthermore, this sequence consists of five independent rational points in $C(\mathbb Q)$. In particular, the rank $r$ of $C(\mathbb Q)$ satisfies $r\ge 5$.