I found this problem in a number theory course, I am assuming (but not sure) it is supposed to be an application of Hensel's lemma.
For every $n \in \mathbb{N}_0$, determine the number of solutions over $\mathbb{Z}/5^n\mathbb{Z}$ of the equation $y^2 = x^3 + x^2 + 5$.
In the base case $n = 1$, there are four solutions $(0, 0), (3, 1), (3, 4), (4, 0)$. The problem with Hensel's lemma in the multivariate case seems to be, if you lift your root to a higher prime power, you can no longer be sure this new root is unique. So instead I figured for every root, I'd fix each variable separately, and consider the univariate problem in the other respective variable. From this I derived the following results:
- $(0, 0)$ doesn't have higher power solutions by fixing either variable.
- $(3, 1)$ has a unique higher power solution by fixing either variable.
- $(3, 4)$ has a unique higher power solution by fixing either variable.
- $(4, 0)$ has a unique higher power solution by fixing the second variable.
However I have no clue how to further deduce the amount of solutions apart from these implicit ones. Just to get an idea of where I was headed I computed all solutions in the case $n = 2$. This way I did notice there are patterns in the solutions:
- $(0, 0)$ mod $5$, there are no corresponding solutions mod $25$.
- $(3, 1)$ mod $5$, the solutions are $(3 + 5k, 21 - 5k)$ mod $25$ for each $k$.
- $(3, 4)$ mod $5$, the solutions are $(3 + 5k, 4 + 5k)$ mod $25$ for each $k$.
- $(4, 0)$ mod $5$, the solutions are $(19, 5k)$ mod $25$ for each $k$.
I can sense there must be a connection between the results I got in the univariate case, and the results I have to get in the multivariate case. But I can't seem to figure out how to formally argue, based on the results of Hensel's lemma, why those patterns in the solutions of $n = 2$ turned out the way they are. Neither do I have a clue how to generalize my results towards an arbitrary power of $5$.