Modified Farey Trees and Pythagorean Triples

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In 1963, F. J. M. Barning discovered a ternary tree of primitive Pythagorean triples, where each triple is transformed to other three triples by three distinct 3 × 3 unimodular matrices. This fact has been rediscovered many times. In this paper, we shall give an elementary explanation of this fact using classical Euclidean parametrization of primitive Pythagorean triples and modified ternary Farey trees.

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