Is it known if $\pi + e$ is transcendental over the rational numbers?

It's not known, and so far as I know nobody even has a reasonable plan of attack on the problem. The whole subject of transcendental number theory is just completely intractable with current machinery.


You might want to consider the following two irrational numbers:

$$\sqrt{2}=1.414213562373095048801688724209698.....$$

$${23481838282\over 245689351}-\sqrt{2}=94.1611061917175085769126386338...$$

Just from a quick look at those two (apparently random) decimal expansions, would you have guessed that the sum of these two irrational numbers is equal to the rational number $23481838281/245689351$?

Does your proof rule out a similar relation between $\pi$ and $e$? How so?

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Pi

Exponential Function

Transcendental Numbers

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