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July 2015 On the distribution of $\tau$-congruent numbers
Chad Tyler Davis, Blair Kenneth Spearman
Proc. Japan Acad. Ser. A Math. Sci. 91(7): 101-103 (July 2015). DOI: 10.3792/pjaa.91.101

Abstract

It is known that a positive integer $n$ is the area of a right triangle with rational sides if and only if the elliptic curve $E^{(n)}: y^{2} = x(x^{2}-n^{2})$ has a rational point of order different than 2. A generalization of this result states that a positive integer $n$ is the area of a triangle with rational sides if and only if there is a nonzero rational number $\tau$ such that the elliptic curve $E^{(n)}_{\tau}: y^{2} = x(x-n\tau)(n+n\tau^{-1})$ has a rational point of order different than 2. Such $n$ are called $\tau$-congruent numbers. It is shown that for a given integer $m>1$, each congruence class modulo $m$ contains infinitely many distinct $\tau$-congruent numbers.

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Chad Tyler Davis. Blair Kenneth Spearman. "On the distribution of $\tau$-congruent numbers." Proc. Japan Acad. Ser. A Math. Sci. 91 (7) 101 - 103, July 2015. https://doi.org/10.3792/pjaa.91.101

Information

Published: July 2015
First available in Project Euclid: 30 June 2015

zbMATH: 1346.14085
MathSciNet: MR3365403
Digital Object Identifier: 10.3792/pjaa.91.101

Subjects:
Primary: 14H52
Secondary: 11G05

Keywords: $\tau$-congruent number , Elliptic curve

Rights: Copyright © 2015 The Japan Academy

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Vol.91 • No. 7 • July 2015
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