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Difference between revisions of "Pythagorean numbers"

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Triplets of positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759501.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759502.png" />. Any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759503.png" /> to this equation, and consequently all (possibly after switching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759504.png" /> only) Pythagorean numbers, can be expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759507.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p0759509.png" /> are positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075950/p07595010.png" />. The Pythagorean numbers can be interpreted as the sides of a right-angled triangle (cf. [[Pythagoras theorem|Pythagoras theorem]]).
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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|Pythagoras theorem]]).
  
  

Revision as of 21:55, 11 April 2014

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).


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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://encyclopediaofmath.org/index.php?title=Pythagorean_numbers&oldid=31584
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article