## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### On the distribution of $\tau$-congruent numbers

#### 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

#### 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

#### References

• M. A. Bennett, Lucas' square pyramid problem revisited, Acta Arith. 105 (2002), no. 4, 341–347.
• J. S. Chahal, On an identity of Desboves, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 3, 105–108.
• J. S. Chahal, Congruent numbers and elliptic curves, Amer. Math. Monthly 113 (2006), no. 4, 308–317.
• A. Desboves, Mémorie sur la Résolution en nombers entiers de l' équation $aX^{m} + bY^{m} = cZ^{n}$, Nouv. Ann. Math., Sér. II 18 (1879), 481–489.
• E. H. Goins and D. Maddox, Heron triangles via elliptic curves, Rocky Mountain J. Math. 36 (2006), no. 5, 1511–1526.
• C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Der Wissenschaften Phys.-math. Kl. 1 (1929), 209–266.