[FOM] a real number for the Ackermann function
Ron Graham
graham at ucsd.edu
Fri Apr 14 22:57:48 EDT 2006
This real is well known to be e^(1/e) where e is the base of the natural
logarithm.
Ron Graham
Robert M. Solovay wrote:
>Do you have some other, more transparent, characterization of this real?
>
> --Bob Solovay
>
>
>On Thu, 13 Apr 2006, Andreas Weiermann wrote:
>
>
>
>>There might be some interest in the
>>following sharp phase transition result
>>for the Ackermann function
>>in terms of a specific (possibly transcendental) real number.
>>
>>Theorem: Let a>1 be a real number. Define a hierarchy of unary functions
>>a_k as follows. Let a_0(x) be a tower of a's of hight x where x is a natural
>>number argument. Let a_{k+1}(x) be the x-th iterate of a_k applied to x.
>>Finally define the diagonal function a(x)=a_x(x).
>>Then x\mapsto a(x) is Ackermannian iff a>1.44466781...
>>
>>Best regards,
>>Andreas Weiermann
>>
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>>
>>
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