Find all pairs of $(n,k)$ where $n ≥ 2$ and $k$ is a prime number such that $log_n (n + k)$ is rational I found the following math question interesting:

Find all pairs of (n,k) where n ≥ 2 and k is a prime number such that log_n (n + k) is rational.

I tried this by contradicting certain statements such as:
log_n (n+k)=2  and  log_n (n+k)=C

I contradicted both (final contradiction shown below):
k=n(n-1)             -- Contradiction since k is a prime number (except n=2)

k=(n^(C-1))(n-1)     -- Contradiction again since k is a prime number

However, I don't know if I did them correct nor how to go from here. What would be the next steps? Are my contradictions correct? If possible can you please explain how you would attack this question with a possible solution?
Thanks a lot.

Let $A=n^p=(n+k)^q$ where $p,q$ are coprime.
Since $A$ is both a $p$-th and $q$-th power, $A=a^{pq}$ where $a$ is an integer. Can you now express $k$ in terms of $a$?
