TY - JOUR

T1 - A unique pair of triangles

AU - Hirakawa, Yoshinosuke

AU - Matsumura, Hideki

N1 - Funding Information:
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”.
Funding Information:
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”.
Publisher Copyright:
© 2018 The Authors

PY - 2019/1

Y1 - 2019/1

N2 - 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.

AB - 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.

KW - Diophantine geometry

KW - Hyperelliptic curves

KW - Rational triangles

UR - http://www.scopus.com/inward/record.url?scp=85054084213&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2018.07.007

DO - 10.1016/j.jnt.2018.07.007

M3 - Article

AN - SCOPUS:85054084213

VL - 194

SP - 297

EP - 302

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -