TITLE: What are some of the implications of $\pi + e$ being rational?
QUESTION [7 upvotes]: Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?

REPLY [7 votes]: One consequence would be that $e \pi$ is transcendental because, for any  $z$,  if $A=e+z$ and $B=ez$ are both algebraic then $e$ is a solution of $x^2-A x+B=0$, which makes $e$ algebraic. But it isn't.