# Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem.

We know that ${\sqrt2}^{\sqrt2}$ is a transcendental number by the Gel'fond-Schneider's theorem. I've tried to prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem, but I'm facing difficulty. I need your help.

This question has been asked previously on math.SE without receiving any answers.

• This is the relevant MSE thread. – Andrés E. Caicedo Jul 31 '13 at 14:40
• You want something a bit more precise. For example, I expect you do not want to deduce this from Kuzmin's result preceding Gel'fond-Schneider. – Andrés E. Caicedo Jul 31 '13 at 14:42
• @Andres Caicedo:Thank you very much for good information. As you wrote, the answer on your page is not what I want. – mathlove Jul 31 '13 at 14:55
• What leads you to expect that this should be possible? There's only one context in which I've seen a discussion of proving something about $\sqrt 2^{\sqrt 2}$ without using Gelfond-Schneider; see math.hmc.edu/funfacts/ffiles/30002.3-5.shtml for example. – Timothy Chow Jul 31 '13 at 15:28
• $\left(\sqrt2^\sqrt2\right)^2=2^\sqrt2$ appears to be irrational, and it looks like an easier thing to prove... – Wlodek Kuperberg Jul 31 '13 at 15:43

• @GerryMyerson: All that is not needed if you deal with the function $2^x$, and only that function is needed for the question. I think the whole proof would just be 2 pages long. The idea is very straightforward (I had that theorem on my oral qual exam for graduate school 35 years ago). – user6976 Aug 1 '13 at 3:16
• @Mark Sapir. A brilliant proof and a clear flow of ideas. The only problem is that almost every particular formula has a misprint in it, starting with $k<r$, which should be $k<n$, and ending with the outlandish choice of $R$ at the culmination moment. If you could correct all these stupid typos that severely restrict the list of potential readers of your nice opus, I'll make all graduate students in my class read it :-). – fedja Sep 5 '13 at 1:08