# [FOM] interesting real number

A. Mani a_mani_sc_gs at yahoo.co.in
Mon Apr 17 18:06:03 EDT 2006

On Sunday 16 April 2006 02:00, Martin Davis wrote:
> Replying to a query by Bob Solovay, Ron Graham wrote:
>  > This real is well known to be e^(1/e)
>
> Just to comment that this number has a nostalgic interest for me. When I
> was a freshman I managed to prove that the sequence defined by s_1=x,
> s_{n+1} = x^(s_n) converges precisely for real numbers x satisfying (1/e)^e
> <= x <= e^{1/e).
>
Are these types of results understood enough to provide a classification
theory . The appropriate starting point may be this general definition :
x is a number of type ... whenever,

If S_{1}=x, S_{n+1}= f (x, S_{n}) for a function f algebraic w.r.t x
and (S_{n}) is convergent  if and only if

\exists a \exists terms  u, v   u(a) =< x =< v(a) and u,v have the same
operation symbols that are repeated the same number of times and u(a), v(a)
are in reduced form.

The bounds are regular in an algebraic sense.

A. Mani
Member, Cal. Math. Soc
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