I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as something which is very easy to prove, in my head I could figure out the rough links quite quickly. I'm not necessarily asking $\textit{for}$ a proof, just confirmation that it's not actually some mathematical mystery that people have been trying to solve for centuries, as that comment implied.
EDIT: Okay I've been humbled, there was a gap in my logic, I'd assumed that $\mathbb{Q}(\pi) \neq \mathbb{Q}(e)$, and then realised after reading all the skepticism that that may not be a simple thing to prove……. after a good amount of time spent over it, I realise that it really, really isn't.
I also admit that this sounded really cocky as a question, sorry. I genuinely believed that it couldn't be as difficult a question as the comment indicated, it didn't seem like it could possibly be an open problem.
Best Answer
It's not known, and so far as I know nobody even has a reasonable plan of attack on the problem. The whole subject of transcendental number theory is just completely intractable with current machinery.