Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the distribution of $\tau$-congruent numbers

Chad Tyler Davis and Blair Kenneth Spearman

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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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 7 (2015), 101-103.

Dates
First available in Project Euclid: 30 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1435669941

Digital Object Identifier
doi:10.3792/pjaa.91.101

Mathematical Reviews number (MathSciNet)
MR3365403

Zentralblatt MATH identifier
1346.14085

Subjects
Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
Elliptic curve $\tau$-congruent number

Citation

Davis, Chad Tyler; Spearman, Blair Kenneth. On the distribution of $\tau$-congruent numbers. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 7, 101--103. doi:10.3792/pjaa.91.101. https://projecteuclid.org/euclid.pja/1435669941


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