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Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?

Marcus M
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PyRulez
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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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    It 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/why-is-it-hard-to-prove-whether-pie-is-an-irrational-number). You can see more [here](https://en.wikipedia.org/wiki/Schanuel's_conjecture) – GAVD Sep 17 '15 at 03:08
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    Possible duplicate of [What would change in mathematics if we knew $\pi+e$ is rational?](https://math.stackexchange.com/questions/730117/what-would-change-in-mathematics-if-we-knew-pie-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
  • @ 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