I've been thinking about pi^e lately, and trying to prove that it is irrational. By rewriting e as 1+1+1/2+1/3!+...+1/n! I got it to pi^2*pi^(1/2)*pi^(1/3!)*...*pi^(1/n!), and proved that each of these terms is irrational. I'm stuck when it comes to showing that multiplied together these numbers are irrational. Any ideas?

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Although we know a lot about certain forms, proving numbers irrational is generally a very, very hard thing to do. For example, I think we don't even know whether or not pi+e is irrational!

No, we do not know if pi+e is irrational, but I don't think that one is very interesting. I think it would be interesting to know if γ(gamma) is irrational though.