You will be familiar with Pythagoras Theorem for right angled triangles:What you may not realise is that it is possible to derive simple expressions to generate all the sets of integer solutionsIt has been known for a long time that it was possible to generate a sequence of triples using formulae. The Pythagoreans were the first to produce such formulae:

Later Euclid gave a formula for producing all triples, withandwhereare positive numbers withandhave no common factors andis odd,is even or vice versa. Theis included as a factor because a right angled triangle can scale up while still remaining a right angled triangle.

Some sets of Pythagorean triples are given below

3 |
4 |
5 |

8 |
6 |
10 |

15 |
8 |
17 |

24 |
10 |
26 |

I chose the above sets for a reason. For each of a, b and c we can find the rule that generates the sequence:

andwith

5 |
12 |
13 |

12 |
16 |
20 |

21 |
20 |
29 |

32 |
24 |
40 |

For the table aboveandwith

By continuing like this we can derive the general formulaeand and includingas the scaling factor gives the formulae above attributed to Euclid.