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