Ref: Professor Stewart’s Cabinet of Mathematical Curiosities: Ian Stewart
Pythagorean Triples:
I can’t really get away without telling you Diophantus’s method for finding all Pythagorean triples, can I?
OK, here it is. Take any two whole numbers, and form:
- twice their product
- the difference between their squares
- the sum of their squares
Then, the resulting three numbers are the sides of a Pythagorean triangle.
For instance, take the numbers 2 and 1. Then,
- twice their product =
- the difference between their squares =
- the sum of their squares =
and, we obtain the famous 3-4-5 triangle. If instead we take numbers 3 and 2, then
- twice their product =
- the difference between their squares =
- the sum of their squares =
and, we get to the next-most-famous 5-12-13 triangle. Taking numbers 42 and 23, on the other hand, leads to
- twice their product =
- the difference between their squares =
- the sum of their squares =
and no one has ever heard of the 1235-1932-2293 triangle. But these numbers do work:
There’s a final twist to Diophantus’s rule. Having worked out the three numbers, we can choose any other number we like and multiply them all by that. So, the 3-4-5 triangle can be converted to a 9-12-15 triangle by multiplying all three numbers by 3, or to an 18-24-30 triangle by multiplying all three numbers by 6. We can’t get these two triples from the above prescription using whole numbers, Diophantus knew that.
More later,
Nalin Pithwa
