The key connecting idea
Henzel's lemma. Let \(f(x)\) be a polynomial with integer coefficients, and let \(m\) and \(k\) be positive integers with \(m \leq k\). If \(r\) is an integer such that \[ f(r) \equiv 0\mod p^k \text{ and } f'(r) \not\equiv 0\mod p \] then there exists an integer \(s\) such that \[ f(s) \equiv 0\mod p^{k+m} \text{ and } r\equiv s \mod p^k .\]
This fact is very helpful to us, because it shows us why a solution to \[ 3r^2 - y \equiv 0 \mod 5 \] (for example) guarantees a solution to \[ 3s^2 - y \equiv 0 \mod 5^k . \]
We can test variations of this! henzel2.c