## Generate Pythagorean Triples

## Answers to Questions

### What is a Pythagorean Triple? (Definition)

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 \( 3^2+4^2=5^2 \)

### How to find a Pythagorean Triple?

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 \( \{3,4,5\} \).

### What is the list of Pythagorean Triples?

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) |

### Is there a right-angled isosceles triangle with integer sides?

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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