Is Gamma Irrational? Investigating the Irrationality of Pi and e

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Discussion Overview

The discussion centers around the irrationality of certain mathematical constants, specifically pi, e, and gamma (γ). Participants explore various approaches to proving the irrationality of these numbers, particularly focusing on pi raised to the power of e and the implications of their sums and products.

Discussion Character

  • Exploratory, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant proposes a method to prove that pi^e is irrational by rewriting e in terms of its series expansion and expressing pi^e as a product of irrational terms.
  • Another participant notes the general difficulty of proving irrationality for numbers, mentioning that it is still unknown whether pi + e is irrational.
  • A different viewpoint challenges the initial approach by suggesting that it is possible to form a rational number from a series with all irrational partial sums.
  • One participant expresses interest in the irrationality of gamma (γ), indicating a shift in focus from pi and e.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the methods for proving irrationality, and multiple competing views remain regarding the approaches to take and the significance of the numbers discussed.

Contextual Notes

There are limitations regarding the assumptions made in the proposed methods, particularly concerning the nature of series and their convergence properties. The discussion also highlights the unresolved status of the irrationality of pi + e.

Who May Find This Useful

Readers interested in number theory, particularly in the properties of irrational numbers and mathematical proofs related to constants like pi, e, and gamma.

Pjennings
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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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