Re: e^(e^(e^e)) =N ?

Phil Carmody a écrit :
"Stephen J. Herschkorn" <sjherschko@xxxxxxxxxxxx> writes:
Phil Carmody wrote:

lundslaktare@xxxxxxxxx a écrit :

PROBLEM FOR COMPUTER SCIENTIST:

Decide if e^(e^(e^e)) is a natural number or not.

e^(e^(e^e)) ~= 2.33150438250040388E1656520


Generalised problem:

Define E_0 = 1; E_{n+1} = e^(E_n)
and A_n = E_n - floor(E_n)

What's A_n's density in [0,1)?
Or:
Does there exist a positive n such that A_n = 0?

That was an intermediate question in my mind before I further expanded to the question that I did ask. Why ask about 1 value,
when you can ask about them all!

Phil


May I conjecture that it's to have two different conjectures :

Carmody's conjecture : 'A_n density in [0,1) is uniform'

Herschkorn (Lundslaktare-Herschkorn?) conjecture : 'there is no positive n such that A_n=0'

(with possible generalization to other irrational values>1 than e?)


These (conjectured, correct me if I'm wrong!) conjectures could hold for a very long time! :-)
Raymond
.

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