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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.

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I'm not sure where you got this problem. It seems to me that it is either impossible or very difficult to get a complete answer.

In particular, your claim that "there are no more solutions" is wrong.

As an example: $a = 2$ and $b = 386$.


There are other families of examples: take any odd $a$ and take $b = a^n$, where $n$ divides $a^{n - 1}$ (e.g. $n = a^k$ for some $k$).

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  • $\begingroup$ I claimed that I was unable to find any more solutions, as opposed to there being no more. But thank you for showing that particular solution, I had not found that. $\endgroup$ – J-S Dec 26 '19 at 17:09
  • $\begingroup$ FYI: even for $a = 2$ there are more: $b = 20136, 59140, 373164544$. See oeis.org/… $\endgroup$ – WhatsUp Dec 26 '19 at 17:11
  • $\begingroup$ Also, for $a = 3$ there are $b = 27, 19683$. This is not yet on OEIS. $\endgroup$ – WhatsUp Dec 26 '19 at 17:14
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There are infinitely many solutions of the following form.

If $p$ is an odd positive integer and $k = p^{p^r}$, then $k^p+1 = p^{p^{r+1}}+1$ divides $p^k+1 = p^{p^{p^r}}+1$. This is because $x+1$ divides $x^p+1$.

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  • $\begingroup$ I don't know why, but I already gave this example (in a more general form) in my answer. $\endgroup$ – WhatsUp Dec 26 '19 at 17:37

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