On sequences of consecutive squares on elliptic curves

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 $(x_i,y_i)\in C(\mathbb{Q}),i=1,2,\dots ,$ is said to form a sequence of consecutive squares on $C$ if the sequence of $x$-coordinates, $x_i,i=1,2,\dots$, 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 $r\ge 5$.

Cited by in corpus (1)

• On geometric progressions on hyperelliptic curves