Professor Andy Hone of the School of Mathematics, Statistics and Actuarial Science (SMSAS) will give an invited virtual talk at the 3rd International Conference on Integrable Systems & Nonlinear Dynamics on Friday 8 October 2021. Professor Jing Ping Wang, also of SMSAS, is part of the organising committee for the conference.

Professor Hone will give a talk on ‘Heron triangles with two rational medians and Somos-5 sequences’.

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

A preprint of this work is available here:

https://arxiv.org/abs/2107.03197