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A Theorem of Fermat on Congruent Number Curves

Abstract : A positive integer $A$ is called a \emph{congruent number} if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a \emph{congruent number} if and only if the congruent number curve $y^2 = x^3 − A^2 x$ has a rational point $(x, y) \in {\mathbb{Q}}^2$ with $y \ne 0$. 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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Submitted on : Wednesday, January 16, 2019 - 12:16:31 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM

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Lorenz Halbeisen, Norbert Hungerbühler. A Theorem of Fermat on Congruent Number Curves. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2019, Hardy-Ramanujan Journal, Atelier Digit_Hum, pp.15 -- 21. ⟨10.46298/hrj.2019.5101⟩. ⟨hal-01983260⟩

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