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Heron triangles with two rational medians and Somos-5 sequences

Hone, Andrew N.W. (2022) Heron triangles with two rational medians and Somos-5 sequences. European Journal of Mathematics, 8 (4). pp. 1424-1486. ISSN 2199-675X. E-ISSN 2199-6768. (doi:10.1007/s40879-022-00586-w) (KAR id:89745)


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 × 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.

Item Type: Article
DOI/Identification number: 10.1007/s40879-022-00586-w
Projects: Cluster algebras with periodicity and discrete dynamics over finite fields, Generalized cluster structures and tau functions for birational difference equations
Additional information: For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
Uncontrolled keywords: Research Article, Heron triangle, Somos sequence, Elliptic function, QRT map, 11D09, 11D25, 33E05, 39A36
Subjects: Q Science > QA Mathematics (inc Computing science)
Q Science > QA Mathematics (inc Computing science) > QA101 Arithmetic
Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Q Science > QA Mathematics (inc Computing science) > QA440 Geometry
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (
Royal Society (
Depositing User: Andrew Hone
Date Deposited: 12 Aug 2021 15:54 UTC
Last Modified: 27 Feb 2024 11:21 UTC
Resource URI: (The current URI for this page, for reference purposes)

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