I'm almost done proving that for any odd numbers $s, t : s > t \geq 1, \ \ \gcd(s, t) = 1$, one can generate PPTs (Primitive Pythagorean Triples) with the formula $$\displaystyle (a, b, c) = \left(st, \frac{s^2-t^2}{2}, \frac{s^2+t^2}{2}\right)$$
However, the only piece I'm missing, is to show that these three expressions, given the initially stated conditions on $s$ and $t$, will be coprime.
I.e. I need to show that $\gcd\left(st, \frac{s^2-t^2}{2}, \frac{s^2+t^2}{2}\right) = 0$.
The book hints at using the fact that if a prime divides a product, it divides at least one factor in the product. In other words $p \mid xy \Rightarrow p\mid x \vee p \mid y$. But I haven't been able to piece that together into a proof.
I've tried a proof by contradiction, where using the definition of divisibility, I tried assuming that there exists a prime $p$ that divides both $st$ and $\frac{s^2- t^2}{2}$, and seeing if I could get a blatant contradiction, but I haven't seen any yet. Or at least not noticed.
I'd especially favor any tip/solution that involves my idea of a proof by contradiction, as it will at least validate some of the efforts I've made so far.
But any help would be appreciated, of course!
We know $p=2$ cannot divide all the factors because $s$ and $t$ are stipulated to be odd, so $2 \nmid st$.
Let $p$ be an odd prime. Suppose $p|st$. By the fact you mentioned, $p|s$ or $p|t$, and not both because $s$ and $t$ are coprime; let's say $p \mid s$ and $p \nmid t$. Then $p \nmid (s+t)$ and $p \nmid (s-t)$, so $p \nmid (s+t)(s-t) = s^2 - t^2$, and thus $p \nmid \frac{s^2-t^2}{2}$.