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.
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14I siuppose the most interesting consequences would be that the methods used in the proof advanced our understanding. – Hagen von Eitzen Sep 23 '16 at 19:25
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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.
Robert Israel
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1There are lots of conjectures that we don't know how to prove, but most experts in the field are pretty confident of. – Robert Israel Sep 23 '16 at 22:40
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2Why they are "pretty confident" if can't prove it. I simply don't understand. Really. – Canis Lupus Sep 24 '16 at 08:37
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So from Schanuel's conjecture, would $\pi e$, $\pi^e$, $pi^\pi$, $e^e$, and $e^\pi^2$ also be transcendental? – Mar 25 '19 at 16:09