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- W4296562785 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 $E(mathbb{Q})$ with Mordell-Weil group $mathbb{Z}times mathbb{Z}/2mathbb{Z}$, 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." @default.
- W4296562785 created "2022-09-21" @default.
- W4296562785 creator A5064714239 @default.
- W4296562785 date "2021-07-07" @default.
- W4296562785 modified "2023-09-26" @default.
- W4296562785 title "Heron triangles with two rational medians and Somos-5 sequences" @default.
- W4296562785 doi "https://doi.org/10.48550/arxiv.2107.03197" @default.
- W4296562785 hasPublicationYear "2021" @default.
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