I would like to prove or disprove the following statement:
There exist integers $m$ and $n$ such that $e = m\pi + n$ (where $e=2.7...$)
Edit: any pointers would be appreciated.
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2Related: Are $\pi$ and $e$ algebraically independent? (though this is much stronger statement than here) – Winther Feb 27 '18 at 23:00
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Thanks for that! So it seems proving my statement will be elusive, but I'm still curious if anyone knows how to disprove it. – J-J Feb 27 '18 at 23:06
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4That's great that you want to take on that challenge. Why post here with the expectation that we prove it for you? "I want to prove this: xyz". That's not even a question. You want to do so, that's nice and all. Now what's your question? – amWhy Feb 27 '18 at 23:12
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1Closely related: Do $e$ and $\pi$ satisfy a non-trivial diophantine equation? The argument there shows this is an open problem. – hardmath Feb 28 '18 at 06:32
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Given the strong regularity of the continued fraction of $e$ and the apparent irregularity of the continued fraction of $\pi$ it looks extremely unlikely that $(\pi\mathbb{Z}+e\mathbb{Z})\cap\mathbb{Z}\neq {0}$. On the other hand we don't even know the exact irrationality measure of $\pi$, so I guess this question is fated to be an open problem for some time. – Jack D'Aurizio Feb 28 '18 at 15:14
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Nobody knows whether $e \pm \pi$ is rational. So it seems most probable that no one knows the answer to your question.
lhf
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