Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?
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Interesting question. I don't have much of a constructive observation to make, except that any of the consequences obviously cannot be things we know to be false, since that would decide the irrationality of $\pi+e$. – Brian Tung Sep 17 '15 at 01:45
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1It would settle some open problems about the approximability of $\pi$ by rationals. – André Nicolas Sep 17 '15 at 01:48
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@BrianTung Well it could, it would just mean its not open anymore. Open problems have been solved on stack exchange before. – Christopher King Sep 17 '15 at 01:55
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Someone asked before, see here. You can see more here – GAVD Sep 17 '15 at 03:08
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2Possible duplicate of What would change in mathematics if we knew $\pi+e$ is rational? – Sil Sep 20 '18 at 19:25
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One consequence would be that $e \pi$ is transcendental because, for any $z$, if $A=e+z$ and $B=ez$ are both algebraic then $e$ is a solution of $x^2-A x+B=0$, which makes $e$ algebraic. But it isn't.
DanielWainfleet
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If $e \pi$ is transcendental, what would the implications be for that? – Frank Bryce Nov 30 '16 at 17:38
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@ John Carpenter . I dk But it might mean the discovery of a technique for answering Q's like it, or perhaps the discovery of new class of numbers. One of David Hilbert's famous problem-set was the nature of $2^{\sqrt 2}$ and numbers like it, which led to the Gelfond Theorem.... – DanielWainfleet Dec 05 '16 at 09:31