I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.


In this question I'm interested to know if there exist a integer $n$ for which the difference $\pi^n-\ e ^n$ is an integer number.


**Note:** even now I got an approach values of the difference $\pi^n-\ e ^n$ for $n=6 $ using [wolfram alpha](https://www.wolframalpha.com/input/?i=pi%5E6-e%5E6&wal=header) which it is closed to $558$ . 


Thank you for any help



Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number?
The claim that $\pi$ and $e$ are known to be algebraically independent is incorrect, see for example [this MO question](https://mathoverflow.net/questions/33817/work-on-independence-of-pi-and-e).


The rationality of $\pi^n-e^n$ is a well-known open problem alredy for $n=1$, and there's no reason to suspect that the $n>1$ case should be any easier.


Not sure if you can find a direct reference for the problem with arbitrary $n$ (for the reason above), but if the result was known it would be a huge deal, and it would be in about every survey on transcendence theory (since there's not that many known results to brag about).


