Halbeisen , Lorenz and Hungerbühler , Norbert - A Theorem of Fermat on Congruent Number Curves

hrj:5101 - Hardy-Ramanujan Journal, January 23, 2019
A Theorem of Fermat on Congruent Number Curves

Authors: Halbeisen , Lorenz and Hungerbühler , Norbert

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.


Source : oai:HAL:hal-01983260v1
Published on: January 23, 2019
Submitted on: January 23, 2019
Keywords: Congruent numbers,Pythagorean triple,2010 Mathematics Subject Classification. primary 11G05; secondary 11D25, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]


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