## Pythagorean Triples

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 solutions It 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, with and where are positive numbers with and have no common factors and is odd, is even or vice versa. The is 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: and with    5 12 13 12 16 20 21 20 29 32 24 40

For the table above and with By continuing like this we can derive the general formulae and and including as the scaling factor gives the formulae above attributed to Euclid. 