Finding all prime powers of the form $\frac{n(n-1)}{10}$

Question

I was trying to come up with a problem where the solutions would be the numbers 5, 6, 10 and 11. It seems like "finding all integers n, where $\frac{n(n-1)}{10}$ is a prime power", is a good attempt, as there are at least no other solutions for n < 1.000.000 (tested with python).

However, I didn't manage to prove that there are no other numbers that match this criteria, so I wonder whether there are none, or whether it's just that prime powers are quite rare, and there could be very large numbers still matching my criteria. Can anyone help me out there?

Answer 1

Hint Since $n - 1$ and $n$ are coprime, at least one of them must be a factor of $10$, hence $n \leq 11$.