TITLE: What would be some consequences if $\pi + e$ was proven to be algebraic vs transcendental over $\Bbb Q$?
QUESTION [6 upvotes]: What would be some consequences if $\pi + e$ was proven to be algebraic vs transcendental over $\Bbb Q$?  I just read that this is an open problem and I wanted to see what some farther reaching consequences would be if this were proven to be algebraic or transcendental.

REPLY [8 votes]: If it were algebraic, that would disprove Schanuel's conjecture.  Which would be a shame, because lots of nice things follow from Schanuel's conjecture.  However,  it's almost certain that $\pi + e$ is transcendental, even though we can't prove it, so I wouldn't worry too much about that possibility.