How do we know that there are infinitly many Pythagorean triples?
In: 2
Depends what you count as a different Pythagorean triple.
For example, the simplest one is 3,4,5.
But from 3,4,5 we also know that 6,8,10 will be one. And 9,12,15, and 12,16,20 and so on. Just knowing *one* Pythagorean triple we can generate an infinite number.
There are also some formulae for generating Pythagorean triples. Euclid’s formula says that taking any two whole numbers, m > n > 0, we can get a Pythagorean triple:
> m^2 – n^(2), 2mn, m^2 + n^2
(a bit of algebra will show those numbers work).
As there are an infinite number of choices for m and n, that formula gives us an infinite number of Pythagorean triples (although there will be overlaps with the 3,4,5 ones).
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