On the distribution of $\tau$-congruent numbers

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Chad Tyler Davis¹  Blair Kenneth Spearman¹ 

[1] Department of Mathematics and Statistics, University of British Columbia Okanagan, Science Building 115, 1177 Research Road

关键词: Elliptic curve;    -congruent number.;   

DOI  :  10.3792/pjaa.91.101

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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.

【 授权许可】

Unknown