[FOM] Some questions regarding irrational numbers

[Timothy Y. Chow]

Lasse Rempe wrote:
> I am not aware of any results excluding the possibility that ln(ln(n)) 
> is an integer for some integer n; but then again I am not sure where I 
> would look for such a theorem.

The first thing to try here is to derive the result from Schanuel's 
conjecture and see if the relevant special case of Schanuel's conjecture 
is a theorem yet.

Schanuel's conjecture says that if a_1, a_2, ..., a_k are linearly 
independent over the rationals, then the transcendence degree of
Q[a_1, a_2, ..., a_k, exp(a_1), exp(a_2), ..., exp(a_k)] over Q
is at least k.  If we take k = 2 and a_1 = n and a_2 = exp(n), then
a_1 and a_2 are linearly independent over the rationals because e
is transcendental.  If Schanuel's conjecture is true, it follows that 
exp(exp(n)) is transcendental.

As far as I know, though, even this special case of Schanuel's conjecture 
is still not known.  Brownawell and Waldschmidt have proved that if a_1 
and a_2 are linearly independent over Q, and b_1 and b_2 are also linearly 
independent over Q, then at least two different numbers amongst a_1, a_2, 
b_1, b_2, exp(a_1 b_1), exp(a_1 b_2), exp(a_2 b_1), exp(a_2 b_2) are 
transcendental.  So for example if we set a_1 = b_1 = 1 and a_2 = b_2 = e 
then it follows that at least one of e^e and e^(e^2) is transcendental.  
But I don't think it's known whether e^e is transcendental.

Tim