Pythagorean numbers

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Triplets of positive integers $x,y,z$ satisfying $x^2+y^2=z^2$. Any solution $x,y,z$ to this equation, and consequently all (possibly after switching $x$ only) Pythagorean numbers, can be expressed as $x=a^2-b^2$, $y=2ab$, $z=a^2+b^2$, where $a$ and $b$ are positive integers $(a>b)$. The Pythagorean numbers can be interpreted as the sides of a right-angled triangle (cf. Pythagoras theorem).



[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII
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This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article