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arXiv · 1602.05862

On sequences of consecutive squares on elliptic curves

Mohamed Kamel·Mohammad Sadek
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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 $r\ge 5$.

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Mohamed Kamel, Mohammad Sadek. 2016-02-18. On sequences of consecutive squares on elliptic curves. https://arxiv.org/abs/1602.05862

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