Is Gamma Irrational? Investigating the Irrationality of Pi and e

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The discussion centers on the irrationality of pi^e, with attempts to prove it by rewriting e as a series and expressing it in terms of pi. The challenge lies in demonstrating that the product of the irrational terms remains irrational. Participants note the general difficulty of proving irrationality, referencing the unresolved question of whether pi+e is irrational. There is a particular interest in the irrationality of gamma (γ), suggesting it could be a more intriguing topic for exploration. The conversation highlights the complexities and ongoing mysteries surrounding irrational numbers in mathematics.
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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!
 
I don't think that is a workable approach, since it's possible to form a rational number as the series where the partial sums are all 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.
 

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