Is Pi multiplied by Euler's Number Rational?

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Open Question

It is not known whether the product of $\pi$ (pi) and Euler's number $e$:

$\pi \times e$

is rational or irrational.



Transcendence of Sum or Product of Transcendentals
Euler's Number is Transcendental
Pi is Transcendental

at least one of $\pi + e$ and $\pi e$ is transcendental.

Also, see Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number.