{"docId":5101,"paperId":5101,"url":"https:\/\/hrj.episciences.org\/5101","doi":"10.46298\/hrj.2019.5101","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01983260","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01983260v1","dateSubmitted":"2019-01-23 14:06:58","dateAccepted":"2019-01-23 15:18:02","datePublished":"2019-01-23 15:19:01","titles":{"en":"A Theorem of Fermat on Congruent Number Curves"},"authors":["Halbeisen\n, \nLorenz","Hungerb\u00fchler\n, \nNorbert"],"abstracts":{"en":"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 $y^2 = x^3 \u2212 A^2 x$ 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."},"keywords":[{"en":"Congruent numbers"},{"en":"Pythagorean triple"},"2010 Mathematics Subject Classification. primary 11G05; secondary 11D25","\n[\nMATH\n] \nMathematics [math]","\n[\nMATH.MATH-NT\n] \nMathematics [math]\/Number Theory [math.NT]"]}