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

1It would settle some open problems about the approximability of $\pi$ by rationals. – André Nicolas Sep 17 '15 at 01:48

@BrianTung Well it *could*, it would just mean its not open anymore. Open problems have been solved on stack exchange before. – PyRulez Sep 17 '15 at 01:55

Someone asked before, see [here](http://math.stackexchange.com/questions/159350/whyisithardtoprovewhetherpieisanirrationalnumber). You can see more [here](https://en.wikipedia.org/wiki/Schanuel's_conjecture) – GAVD Sep 17 '15 at 03:08

2Possible duplicate of [What would change in mathematics if we knew $\pi+e$ is rational?](https://math.stackexchange.com/questions/730117/whatwouldchangeinmathematicsifweknewpieisrational) – 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 $x^2A x+B=0$, 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

@ 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 problemset was the nature of $2^{\sqrt 2}$ and numbers like it, which led to the Gelfond Theorem.... – DanielWainfleet Dec 05 '16 at 09:31