Some variants of the congruent number problem. I
type: Item
instance of: scholarly article
zbMATH Open document ID: 1025.11015
author: Shin-ichi Yoshida
published in: Kyushu Journal of Mathematics
publication date: 2003-08-24T00:00:00Z
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1708488
zbMATH Keywords: congruent numbers
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
full work available at URL: kyushujm.55.387
OpenAlex ID: W4236095906
DOI: 10.2206/KYUSHUJM.55.387
title: Some variants of the congruent number problem. I
review text: Let \(\theta\) be a real number with \(0<\theta <\pi\). A natural number \(n\) is said to be \(\theta\)-congruent if there exists a triangle with rational sides and an angle \(\theta\) whose area is \(n\sqrt{r^2-s^2}\), where \(\cos \theta= s/r\) with \(r,s\in\mathbb{Z}\), \(\text{gcd} (r,s)=1\), and \(r>0\). In this paper, among other results, the author proves the following. For a prime number \(p\), (i) if \(p\equiv 7,11,13\pmod {24}\), then \(p\) is not \(2\pi/3\)-congruent, and (ii) if \(p\equiv-1\pmod {24}\) then \(p,2p\), and \(3p\) are \(2\pi/3\)-congruent.
reviewed by: Fumio Hazama
description: scientific article; zbMATH DE number 1708488