, 2022

Solution:
Rewriting the equation as $p^{2} = m (m^{2} - p)$, we see that $m$ must divide $p^{2}$. However, since $p$ is prime, the only positive factors of $p^{2}$ are $1$, $p$ and $p^{2}$. We now check each case separately:

Case $m = 1$:
The equation becomes $p^{2} + p = 1$. Since this would imply that $p$ divides $1$, we do not get any solutions in this case.

Case $m = p$:
The equation becomes $2 p^{2} = p^{3}$ and cancelling a factor of $p^{2}$ we find that $p = 2$. The pair $(2, 2)$ is therefore the only solution in this case.

Case $m = p^{2}$:
The equation becomes $p^{2} + p^{3} = p^{6}$ and after cancelling, we get $1 + p = p^{4}$. Again, this would imply that $p$ divides $1$, which is not possible. No solutions in this case.

We conclude that $(2, 2)$ is the only pair satisfying the equation.

Solution:
We observe that $p$ divides the left-hand-side of the equation and therefore must divide the right-hand-side as well. Now if $p$ divides $m^{3}$, we must have that $p$ divides $m$. Let us write $m = p n$ for some positive integer $n$. Substituting into our equation and cancelling a factor of $p^{2}$ we are left with $1 + n = p n^{3}$. This implies that $n$ must divide $1$ and therefore $n = 1$ and $m = 2$. The equation now simplifies to $p = 2$ and we conclude that the only solution is the pair $(2, 2)$.