Hi, because of a question on this week's on Brilliant problem set, I've come across the Euclid's formula for generating Pythagorean triples for the first time.
The formula says that (in my own words) for a right angled triangle with sides a,b and hypotenuse c,
\[a^2+b^2 = c^2 \iff (m^2-n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 \]
for any 2 integers \(m > n\). After staring at this formula for a while, I realised it's actually a simple and straightforward case of expansion on LHS and RHS, followed by a check on whether the terms matched on both sides.
This made me wonder if Euclid came up with this theorem because of some eureka moment (you know, like maybe one day he's just sitting alone at his desk and this idea that \( (m^2-n^2) + (2mn)^2 = (m^2 + n^2)^2 \) can be applied to the Pythagoras theorem just pops into his head) or was he driven by any other sort of motivation to derive this formula?
And this brings me to another question, suppose Euclid didn't leave any proof of the derivation of this formula because he thought that it was such a straight forward application of the idea of \( (m^2-n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 \) to \(a^2 + b^2 = c^2\). Would there had been any practical need to come up with a derivation from \(a^2 + b^2 = c^2\)?
I apologize if I sound like an idiot asking such questions, but well...I just couldn't get these questions off my mind :p