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Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces

Robin Hartshorne·Ronald van Luijk
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Abstract

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.

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Robin Hartshorne, Ronald van Luijk. 2006-06-28. Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces. https://arxiv.org/abs/math/0606700

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