I found a question that asked me to discuss the number of solutions of the equation $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$ in function of $k$.
I would like to use the multivariate Lemma of Hensel-Rychlick for $p$-adic numbers. Suppose that $N$ equals the number of solutions of $y^2 = x^3 + x^2 +5$ in $\mathbb{Z}/p\mathbb{Z}$ such that there exist some $i \in \{x, y\}$ such that $\frac{d f(a)}{di} \neq 0$ where $f(a) \equiv 0 \mod p$. For each of these $N$ solutions there exist a $b \in \mathbb{Z}_p$ such that $f(b) = 0$. This essentially means that for each $b$ and $k \in \mathbb{N}_{>0}$, there exists a solution for $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$. So there are $N$ solutions of $y^2 = x^3 + x^2 +5$ in $\mathbb{Z}/p^k \mathbb{Z}$.
What do you guys think of this answer?