eng
episciences.org
Hardy-Ramanujan Journal
2804-7370
2019-01-23
10.46298/hrj.2019.5101
5101
journal article
A Theorem of Fermat on Congruent Number Curves
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.
https://hrj.episciences.org/5101/pdf
Congruent numbers
Pythagorean triple
2010 Mathematics Subject Classification. primary 11G05; secondary 11D25
[ MATH ] Mathematics [math]
[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]