A Theorem of Fermat on Congruent Number Curves

Lorenz Halbeisen; Norbert Hungerbühler

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arXiv · 1803.09604

A Theorem of Fermat on Congruent Number Curves

Lorenz Halbeisen·Norbert Hungerbühler

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Abstract

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.

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Lorenz Halbeisen, Norbert Hungerbühler. 2018-03-26. A Theorem of Fermat on Congruent Number Curves. https://arxiv.org/abs/1803.09604

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