A unique pair of triangles
Number Theory
2018-09-27 v1
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 -adic abelian integrals.
Cite
@article{arxiv.1809.09936,
title = {A unique pair of triangles},
author = {Yoshinosuke Hirakawa and Hideki Matsumura},
journal= {arXiv preprint arXiv:1809.09936},
year = {2018}
}
Comments
5 pages, to appear in Journal of Number Theory, Some modifications are added to the article published online