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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]]).
 
  
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''Pythagorean triple''
  
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Triplets of positive integers $x,y,z$ satisfying the [[Diophantine equations|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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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XIII</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XIII</TD></TR>
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Latest revision as of 07:30, 10 December 2016

2020 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).

Comments

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
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