(x^x)^(x^x), x^(x^(x^x)), etc...

[Edwin Clark]

On Thu, 1 May 2003, benoit wrote:

> 
> And what about the number of integer values taken by :
> x
> x^x,
> x^x^x,
> x^x^x^x
> 
> using parentheses as desired
> 
> when x=sqrt(2) ,  x=2^(1/3) .....

When x=sqrt(2) it appears that the integer values obtained are

2, 4, 16, 256, ..., 2^(2^n), ...

It is easy to see that these are all obtained, but are there 
any others? (Not for n to 10.)

I guess a similar result holds for x = k^(1/k) k=3,4,... Namely, the
integer values obtained are precisely the numbers k^(k^n)
but I cannot prove it. 


> 
> when  x=sqrt(2)  n=3 there  is a well known example :
> 
> (sqrt(2)^sqrt(2))^sqrt(2)=sqrt(2)^(sqt(2)*sqrt(2))=sqrt(2)^2=2 integer
> 
> hence a(1)=0 a(2)=0 a(3)=1 a(4)>=1...
> 

To continue this: Maple yields the following number of distinct
integer values with n x's:

for x=sqrt(2) for n from 1 to 10:

 0, 0, 1, 1, 2, 2, 3, 4, 5, 7

for x = 3^(1/3) for n from 1 to 9:

0, 0, 0, 1, 1, 1, 2, 3, 4

for x = 2^(1/3) for n from 1 to 9:

0, 0, 0, 0, 0, 0, 0, 0, 0


Edwin