## Lorenz Halbeisen ; Norbert Hungerbühler - A Theorem of Fermat on Congruent Number Curves

hrj:5101 - Hardy-Ramanujan Journal, January 23, 2019 - https://doi.org/10.46298/hrj.2019.5101
A Theorem of Fermat on Congruent Number Curves

Authors: Lorenz Halbeisen ; Norbert Hungerbühler

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.

Published on: January 23, 2019
Imported 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]