1. A Theorem of Fermat on Congruent Number Curves
- Author
-
Lorenz Halbeisen and Norbert Hungerbühler
- Subjects
Congruent numbers ,[ MATH ] Mathematics [math] ,Fermat's Last Theorem ,2010 Mathematics Subject Classification. primary 11G05 ,secondary 11D25 ,Mathematics - Number Theory ,Mathematics::General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,Pythagorean triple ,01 natural sciences ,[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,11G05, 11D25 ,Integer ,Rational point ,Elementary proof ,FOS: Mathematics ,Mathematics::Metric Geometry ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,Congruent number - 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., Comment: 8 pages, one figure
- Published
- 2019
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