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Heron triangles and the hunt for unicorns

Hone, Andrew N.W. (2024) Heron triangles and the hunt for unicorns. Mathematical Intelligencer, . ISSN 0343-6993. (In press) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:105015)

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A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be one whose side lengths and area are all rational numbers. A perfect triangle is a Heron triangle with all three medians being rational. According to a longstanding conjecture, no such triangle exists, so perfect triangles are as rare as unicorns. However, if perfect is the enemy of good, then perhaps it is best to insist on only two of the medians being rational. Buchholz and Rathbun found an infinite family of Heron triangles with two rational medians, which they were able to associate with the set of rational points on an elliptic curve E(Q). Here we describe a recently discovered explicit formula for the sides, area and medians of these (almost perfect) triangles, expressed in terms of a pair of integer sequences: these are Somos sequences, which first became popular thanks to David Gale's column in Mathematical Intelligencer.

Item Type: Article
Additional information: For the purpose of open access, the author(s) has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra > QA241 Number theory
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: 16 Feb 2024 16:15 UTC
Last Modified: 27 Feb 2024 10:20 UTC
Resource URI: (The current URI for this page, for reference purposes)

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