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

Question

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.

Answer 1

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.