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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 which it is closed to $558$ .

Thank you for any help

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    $\begingroup$ It is a wide open problem in transcendental number theory to prove the algebraic independence of $e$ and $\pi$. Even proving irrationality of $\pi - e$ has resisted all attempts so far. It doesn't appear to me that proving that $\pi^n - e^n$ is never a rational integer would be any easier than this well known unsolved problem. $\endgroup$
    – Vesselin Dimitrov
    Commented Oct 31, 2016 at 22:58
  • $\begingroup$ sorry for that , i think it's not open $\endgroup$
    – zeraoulia rafik
    Commented Oct 31, 2016 at 23:30
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    $\begingroup$ @user51189. According to your first sentence the polynomial equation $f(X,Y)=X^n-Y^n+m=0$ when substituted by $\pi$ and $e$ will not be satisfied. So the question contradictst your first sentence. If you think a problem is closed, you must be aware someone has settled it and you should make the whole world know about this. $\endgroup$
    – P Vanchinathan
    Commented Nov 1, 2016 at 0:09
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    $\begingroup$ If you think it's not open, then you think the answer is known. Now why would you think that? $\endgroup$
    – Todd Trimble
    Commented Nov 1, 2016 at 0:09
  • $\begingroup$ "close to 558" (i.e. approximately 557.96...)? Sure, for any particular integer nonzero $n$ you can presumably compute $\pi^n - e^n$ numerically with sufficient accuracy to show that this particular one is not an integer. That tells you nothing about whether it will be an integer for some $n$ that you haven't tried yet. $\endgroup$
    – Robert Israel
    Commented Nov 1, 2016 at 1:18

1 Answer 1

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The claim that $\pi$ and $e$ are known to be algebraically independent is incorrect, see for example this MO question.

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).

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