What would be some consequences if $\pi + e$ was proven to be algebraic vs transcendental over $\Bbb Q$? 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.

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.
