Open Access
May 2017 On the distribution of rank two $\tau$-congruent numbers
Chad Tyler Davis
Proc. Japan Acad. Ser. A Math. Sci. 93(5): 37-40 (May 2017). DOI: 10.3792/pjaa.93.37


A positive integer $n$ is the area of a Heron triangle if and only if there is a non-zero rational number $\tau$ such that the elliptic curve \begin{equation*} E_{τ}^{(n)}: Y^{2} = X(X-nτ)(X+nτ^{-1}) \end{equation*} has a rational point of order different than two. Such integers $n$ are called $\tau$-congruent numbers. In this paper, we show that for a given positive integer $p$, and a given non-zero rational number $\tau$, there exist infinitely many $\tau$-congruent numbers in every residue class modulo $p$ whose corresponding elliptic curves have rank at least two.


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Chad Tyler Davis. "On the distribution of rank two $\tau$-congruent numbers." Proc. Japan Acad. Ser. A Math. Sci. 93 (5) 37 - 40, May 2017.


Published: May 2017
First available in Project Euclid: 29 April 2017

zbMATH: 1374.14028
MathSciNet: MR3645658
Digital Object Identifier: 10.3792/pjaa.93.37

Primary: 14H52
Secondary: 11G05

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

Rights: Copyright © 2017 The Japan Academy

Vol.93 • No. 5 • May 2017
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