Proof of uniqueness of Pythagorean triples generated by Euclid's formula

Question

How it can be proven that Euclid's formula for generating Pythagorean triples obtains each$$a=m^2-n^2,\,b=2mn,\,c=m^2+n^2$$at unique $m,\,n$?

Thanks in advance.

Answer 1

A right triangle with side lengths $a,b,c$ where $c$ is the hypotenuse and using integers $m, n $ where $ m > n$, we can find Euclid's Formula.

First, given a right triangle with an hypotenuse $\sqrt{c}$ we know that the side lengths must be m and n telling us $c = m^2 + n^2$.

Using this we now can find the integer values of a and b.

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

$m^4 + 2m^2n^2 + n^4 = c^2$

$(m - n)^2 + (2mn)^2 = c^2$

Which results in Euclid's formula:

$(2mn, m^2 - n^2, m^2 + n^2)$