A Theorem of Fermat on Congruent Number Curves

2018/03/26 by Halbeisen, Lorenz, Hungerbühler, Norbert
#11D25 #11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.09604

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 y²=x³-A²x has a rational point (x,y)∈\mathbb Q² with y≠ 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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