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

In: 2

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).

There are some good proofs already, here is a pretty fun one. First thing to notice is that a Pythagorean Triple is basically the same thing as a point on the unit circle whose coordinates are rational numbers (fractions of whole numbers). For instance, the 3,4,5 Pythagorean triple corresponds to the point (3/5,4/5) on the unit circle and vice versa. The second thing to notice is that we can “add” points together on the unit circle, just start at one and rotate it by the angle of the second. There is a very nice formula for the point that you get in this “addition”, if you start with (a,b) and (c,d), then they add together to make the point (ac-bd,ad+bc), this rotates the first one by the angle of the second. This is basically the angle addition formula for sine and cosine. Something to notice about this is that this formula is very basic and, specifically, if you start with all rational numbers then you’ll get rational numbers back in return. So we can use this to successively add pythagorean triples together to get *new* Pythagorean triples!

Starting with just (3/5,4/5) we can add this to itself multiple times to get

* (3/5, 4/5) corresponding to 3,4,5

* (-7/25, 24/25) corresponding to 7,24,25

* (-117/125, 44/125) corresponding to 44,117,125

And you can go on forever like this, creating infinitely many. Basically, just rotate a known Pythagorean Triple around the circle by the angle it makes and you’ll pick up infinitely many! Start with different triples to get different ones! This is actually used in higher math to show the existence of points on curves. Notably, well beyond the scope of ELI5, the Birch and Swinnerton-Dyer Conjecture tries to identify how many ways we can generate infinitely many points on things called Elliptic Curves. It’s still unsolved (and has a $1million prize attached to a proof of it), but the best partial results for it find infinitely many points by finding a good starting point (analogous to a starting Pythagorean Triple), and then adding it to itself infinitely many times!

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.