Hone, Andrew N.W. (2021) Heron triangles with two rational medians and Somos5 sequences. Proceedings of the London Mathematical Society, . ISSN 00246115. (Submitted) (KAR id:89745)
PDF
Preprint
Language: English 

Download (1MB)
Preview

Preview 
This file may not be suitable for users of assistive technology.
Request an accessible format

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. 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 MordellWeil group Z × Z/2Z, and they observed a connection with a pair of Somos5 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 Somos5 sequences and associated QuispelRobertsThompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.
Item Type:  Article 

Projects:  Projects 26013 not found. Projects 23085 not found. 
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 
Depositing User:  Andrew Hone 
Date Deposited:  12 Aug 2021 15:54 UTC 
Last Modified:  12 Aug 2021 15:54 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/89745 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
 Link to SensusAccess
 Export to:
 RefWorks
 EPrints3 XML
 BibTeX
 CSV
 Depositors only (login required):