I've recently been attempting to solve the following problem and made some initial progress. However, im not able to progress any further, any hint would be greatly appreciated!
Let $a$ and $b$ be positive integers. For which values of $a$ and $b$ is the quotient $\frac{a^b + 1}{b^a + 1}$ an integer?
Thus far I have found that when $b=1$, then $\frac{a^b + 1}{b^a + 1}$ is an integer for all odd positive integers $a$. Likewise, the quotient is an integer when $a=b$ and when $(a,b)=(2,4)$ and $(a,b)=(4,2)$.
I haven't yet been able to find any more solutions and I also have no ideas on how I can prove that there are no more solutions, other than the ones given.