Why is it hard to prove whether $\pi+e$ is an irrational number?

Question

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?"

Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but ultimately failed to conclude this.

Answer 1

"Why is this hard?" I think a different question would be "Why would it be easy?"

But there are some things that are known. It is known that $\pi$ and $e$ are transcendental. Thus $(x-\pi)(x-e) = x^2 - (e + \pi)x + e\pi$ cannot have rational coefficients. So at least one of $e + \pi$ and $e\pi$ is irrational. It's also known that at least one of $e \pi$ and $e^{\pi^2}$ is irrational (see, e.g., this post at MO).