# How do we know that there are infinitly many Pythagorean triples?

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How do we know that there are infinitly many Pythagorean triples?

In: 2 Euclid’s proof:

Consider the identity n²+2n+1 = (n+1)²

Whenever 2n+1 is a square, this forms a Pythagorean triple. But 2n+1 comprises all the odd numbers

Every other square number is odd

There are an infinite number of odd squares

Hence there are an infinite number of Pythagorean triples. Because we construct infinitely many. Pick any two distinct natural numbers n and m. Then we have a pythagorean triple by

a=m^(2)-n^(2)

b=2mn

c=m^(2)+n^(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). 