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 includingas the scaling factor gives the formulae above attributed to Euclid.