# Pythagorean numbers

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 11D09 [MSN][ZBL]

*Pythagorean triple*

Triplets of positive integers $x,y,z$ satisfying the Diophantine equation $x^2+y^2=z^2$. After removing a common factor, and possibly switching $x,y$, any solution $(x,y,z)$ to this equation, and consequently all Pythagorean numbers, can be obtained 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).

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#### References

[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII |

**How to Cite This Entry:**

Pythagorean numbers.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pythagorean_numbers&oldid=39945

This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article