## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 38, Number 2 (2015), 352-364.

### On θ-congruent numbers on real quadratic number fields

Ali S. Janfada and Sajad Salami

#### Abstract

Let
**K** =
**Q** (
) be a real quadratic number field, where
*m* > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) =
*s/r* with 0 < |
*s*| <
*r* and gcd(
*r,s*) = 1. A positive integer
*n* is called a (
**K**,θ)-congruent number if there is a triangle, called the (
**K**,θ,
*n*)-triangles, with sides in
**K** having θ as an angle and
*n*α
_{θ} as area, where α
_{θ} =
. Consider the (
**K**,θ)-congruent number elliptic curve
*E*
_{
n,θ
}:
*y*
^{2} =
*x*(
*x* + (
*r* +
*s*)
*n*) (
*x* − (
*r* −
*s*)
*n*) defined over
**K**. Denote the squarefree part of positive integer
*t* by sqf(
*t*). In this work, it is proved that if
*m* ≠ sqf(2
*r*(
*r* −
*s*)) and
*mn* ≠ 2, 3, 6, then
*n* is a (
**K**,θ)-congruent number if and only if the Mordell-Weil group
*E*
_{
n,θ
}(
**K**) has positive rank, and all of the (
**K**, θ,
*n*)-triangles are classified in four types.

#### Article information

**Source**

Kodai Math. J., Volume 38, Number 2 (2015), 352-364.

**Dates**

First available in Project Euclid: 9 July 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1436403896

**Digital Object Identifier**

doi:10.2996/kmj/1436403896

**Mathematical Reviews number (MathSciNet)**

MR3368071

**Zentralblatt MATH identifier**

06481140

#### Citation

Janfada, Ali S.; Salami, Sajad. On θ-congruent numbers on real quadratic number fields. Kodai Math. J. 38 (2015), no. 2, 352--364. doi:10.2996/kmj/1436403896. https://projecteuclid.org/euclid.kmj/1436403896