TITLE: What would change in mathematics if we knew $\pi+e$ is rational?
QUESTION [22 upvotes]: It is well known that there's no conclusion now whether $\pi+e$ is rational or not. What would happen if we knew that $\pi+e$ is rational?  Specifically, are there related open problems that would be settled?

REPLY [5 votes]: It would imply in particular that $e$ is a period, which it conjecturally isn't. There are deep reasons why $e$ is conjectured not to be a period, stemming (I believe) from Deligne's theory of motivic weights. I recommend taking a look at Kontsevitch and Zagier's fantastic article Periods.