I've used this proof technique to examine what happens when $ax = by$ for example. The proof worked out nicely by introducing the $n$th prime $p$.

Here's exactly where I used it before

What I want from you is either an example where such prime induction proof has also been used, or to explain why you do not see it (at all!).

To motivate this, the application is proving that $U_{a} = \{ ax + k : x \in \Bbb{Z}\}$ forms a basis for a topology on $\Bbb{Z}$, given some fixed $k$. Since then if $U_a \cap U_b \ni z \implies w = ax = by$. Hence the need for the proposition. But also apply it to $V_a = \{ ax^r + k : x \in \Bbb{Z}\}$ such that $r\geq 2$, $k \in \Bbb{Z}$ are fixed.