Tool to generate Pythagorean triples. A Pythagorean triple is a set of three natural integer numbers (a,b,c), such that a^2+b^2=c^2

Pythagore Triple - dCode

Tag(s) : Arithmetics, Geometry

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A Pythagorean triplet is a set of three natural numbers $ a $, $ b $ and $ c $ such that $ a^2+b^2=c^2 $

__Example:__ (3,4,5) is a triplet of Pythagoras because $ 3^2+4^2=5^2 $

It exists heuristics to find a Pythagore Triple but the easiest method consists in testing iteratively all possibilities of a and b when s is given, the value of c is constrained by s=a+b+c.

The following equations can be deducted:

$$ a^2 + b^2 = (s − a − b)^2 \\ a <= (s − 3)/3 \\ b < (s − a)/2 $$

__Example:__ If $ s = 12 $, then $ a <= 3 $ and $ b < 4.5 $, a quick test allows to find $ a = 3, b = 4 $ and get the triple $ \{3,4,5\} $.

Is (X,Y,Z) a Pythagorean triple? Use the checker above to find out. Otherwise, manually, take for a and b the 2 smallest values among X, Y, Z, and for c the largest value then calculate first $ a ^ 2 + b ^ 2 $ then $ c ^ 2 $ if the 2 values found are identical then (X, Y, Z) is a Pythagorean triplet, otherwise it is not a Pythagorean triple.

The first Pythagorean triples (side inferior to 100)

(3,4,5) | (5,12,13) | (6,8,10) |

(7,24,25) | (8,15,17) | (9,12,15) |

(9,40,41) | (10,24,26) | (11,60,61) |

(12,16,20) | (12,35,37) | (13,84,85) |

(14,48,50) | (15,20,25) | (15,36,39) |

(16,30,34) | (16,63,65) | (18,24,30) |

(18,80,82) | (20,21,29) | (20,48,52) |

(21,28,35) | (21,72,75) | (24,32,40) |

(24,45,51) | (24,70,74) | (25,60,65) |

(27,36,45) | (28,45,53) | (28,96,100) |

(30,40,50) | (30,72,78) | (32,60,68) |

(33,44,55) | (33,56,65) | (35,84,91) |

(36,48,60) | (36,77,85) | (39,52,65) |

(39,80,89) | (40,42,58) | (40,75,85) |

(42,56,70) | (45,60,75) | (48,55,73) |

(48,64,80) | (51,68,85) | (54,72,90) |

(57,76,95) | (60,63,87) | (60,80,100) |

(65,72,97) |

There is no Pythagorean triplet with 2 identical values. Indeed if 2 sides are $ a $ (natural integer), the last side is $ a \sqrt2 $ which can not be an integer.

__Example:__ $ a = 1 $ the triplet becomes $ (1, 1, \sqrt2) $. By scaling, it is not possible to obtain an both isosceles and right triangle with integer sides.

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Cite as source (bibliography):

*Pythagore Triple* on dCode.fr [online website], retrieved on 2024-09-09,

pythagore,pythagorean,triple,triangle,integer,coprime,square,hypothenuse

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