VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Heron triangles with two rational medians and Somos-5 sequences

Andy Hone (University of Kent)

Wednesday 28th April 13:30-14:30 Zoom seminar hosted by ICMS

Abstract

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. In fact Schubert made the erroneous assertion that even two rational medians was impossible, but Buchholz and Rathbun later  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+Z, and they observed an apparent 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 and the area in this infinite family of Heron triangles with two rational medians.

 

For more information, and how to access the seminar through Zoom, see the webpage of the events here.

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