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In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each exponent level but different.

I suspect this case is not possible, but haven't been able to formally rule it out. So to make it concrete, is it possible that two power towers $$ S = p^{q^{\Large r^{\Large s^t}}} \quad T = v^{w^{\Large x ^ {\Large y^z}}} $$ with $1<p,q,r,s,t,v,w,x,y,z<N=2^{10}$ satisfy $$ S<T<2S $$ ?

If not and ideally can we bound the distance in terms of $N$? As @fedja commented, without a constraint we can always let $M=q^{r^{s^t}}=w^{x^{y^z}}$ and find $(v/p)^M<2$ with $p$ large enough.

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    $\begingroup$ Certainly. No matter what other variables are, $p^M$ and $v^N$ can have a ratio as close to $1$ as you wish (if nothing better, put $p=k^N$ and $v=(k+1)^M$ and take $k$ really large). $\endgroup$ – fedja Jun 18 '13 at 3:46
  • $\begingroup$ @fedja Thanks. My first formulation of the question was poor, but I'd like assess the viability of an algorithm so I also need a bound on the size of the "inputs." $\endgroup$ – Zander Jun 18 '13 at 4:11
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    $\begingroup$ @Zander It seems that my bounty was not enough to attract a good answer. :( $\endgroup$ – Lord_Farin Nov 4 '13 at 11:59
  • $\begingroup$ @Zander Intuitively, I would guess that the quotient of distinct power towers with $5$ entries greater than $1$, with the larger one being the numerator, must be astronomical (unless the base is itself astronomical) , so I would be extremely surprised if it can be smaller than $2$. Maybe, Deedlit can prove this formally. $\endgroup$ – Peter Mar 21 '17 at 20:28

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