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Am I right in thinking this is not possible since both are known to be transcendental?

Also, $e^{i\pi}+1=0$ suggests this is not possible - we can not isolate $e$ or $\pi$ from this since it involves taking a log at some point, thus "cancelling" $e$.

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This is unknown at the moment. They both are transcendental, but it is unknown if they are algebraically independent. –  M Turgeon Aug 30 '12 at 11:50
    
But doesn't $e^{i\pi}+1=0$ show them to be algebraically independent? –  pbs Aug 30 '12 at 11:55
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No. It simply shows that there exists a non-algebraic relation between them. –  M Turgeon Aug 30 '12 at 11:57
    
So there could be an algebraic relation that exists that we just don't know of. Ok. –  pbs Aug 30 '12 at 11:59
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Note that $e$ and $2e$ are both known to be transcendental, and despite that it is possible to express each in terms of the other. –  Gerry Myerson Aug 30 '12 at 12:50
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The answer is: This is unknown at the moment. They both are transcendental, but it is unknown if they are algebraically independent. If you want to know more about this, you can read this Mathoverflow thread, or this Wikipedia article on Schanuel's conjecture.

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