Can √p√p√p\sqrt{p}^{\sqrt{p}^{\sqrt{p}}} be an integer, if pp is a non-square positive integer?

Can ppp be an integer, when p is a non-square positive integer?

Of course, it seems it would never but is there a proof of the fact, or maybe we have some spooky p that makes it valid?

Answer

Although such a number almost surely can not be an integer, it seems that this should be an open problem still.

However the claim that it can not be an integer would follow from Schanuel’s conjecture:

Given any n complex numbers z1,,zn which are linearly
independent over Q, the extension field
Q(z1,,zn,ez1,,ezn) has transcendence
degree at least n over Q, i.e. the set {z1,,zn,ez1,,ezn} contains at least n algebraically
independent numbers.

This is a strong statement. For example, let z1=1 and z2=iπ. Then the set {1,iπ,e,1} must contain two algebraically independent numbers. Then π and e would be algebraically independent, showing that, for example, e+π and eπ are transcendental (which otherwise is an open problem).

Now consider an algebraic number α which is irrational (such as m for m not a square). By the Gelfond-Schneider theorem, αα is transcendental. Therefore, logα, αlogα, and ααlogα are Q-linearly independent.

By Schanuel’s conjecture, the set logα,αlogα,ααlogα,α,αα,ααα
contains at least three elements that are algebraically independent. But α is algebraic. Also αlogα and ααlogα are algebraically dependent on logα and αα, so logα, αα and ααα must be algebraically independent.

In particular, ααα must be transcendental.

For more on this subject, including more and stronger consequences on Schanuel’s conjecture, please see the papers by Marques and Sondow, Schanuel’s conjecture and algebraic powers zw and wz with z and w transcendental and The Schanuel Subset Conjecture implies Gelfond’s Power Tower Conjecture.

I would be interested if anyone thought that, with this kind of problem, that proving something is not rational or not an integer might be substantially more tractable than proving it is transcendental.

Attribution
Source : Link , Question Author : Sawarnik , Answer Author : John M

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