Irrationality of e+ π and πe

Known. $e$ and $\pi$ are both irrational (indeed transcendental).

If $e+\pi\in\mathbb{Q}$. Then $e,\pi$ are roots of $X^2-sX+p=0$ where $s=e+\pi$ and $p=e\pi$. If both $s,p\in\mathbb{Q}$, then $e,\pi$ are algebraic — contradiction with transcendence.

But only one is known? Actually, if $e+\pi\in\mathbb{Q}$, then $e$ and $\pi$ are roots of $X^2-sX+p=0$ for some $p$. But we do not know $p=e\pi$ is rational.

Open problem. Whether $e+\pi$ and $e\pi$ are irrational is open. (However, at most one of them can be rational, since their sum and product being rational would force both to be algebraic.)

Conclusion. These are famous open problems in transcendence theory.