Elsevier

Journal of Number Theory

Volume 194, January 2019, Pages 297-302
Journal of Number Theory

A unique pair of triangles

https://doi.org/10.1016/j.jnt.2018.07.007Get rights and content
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open access

Abstract

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p-adic abelian integrals.

MSC

primary
14G05
secondary
11G30
11Y50

Keywords

Diophantine geometry
Hyperelliptic curves
Rational triangles

Cited by (0)

This research was supported by JSPS KAKENHI Grant Number JP15J05818 and the Research Grant of Keio Leading-edge Laboratory of Science & Technology (Grant Numbers 000036 and 000053). This research was also conducted as part of the KiPAS program FY2014-2018 of the Faculty of Science and Technology at Keio University, and was supported in part by JSPS KAKENHI 26247004, 18H05233, as well as the JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.