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$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$ Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$.

Are there any positive integer $n$ and negative integer $a$ such that $f_n(a)$ is an integer?

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    $\begingroup$ If $a$ is, say, $-1$, we are asking about $$2^{2^{2^{2^{2^{\sqrt2}}}}}$$ I think the question of whether that (and things like it) is an integer is surely "research level", albeit probably hopeless. Maybe it should be closed, but not "research level"? Voting to reopen. $\endgroup$ Aug 24, 2013 at 1:08
  • $\begingroup$ Why 8 instances of 2? Why not 5 or 6 or 7? $\endgroup$
    – rghthndsd
    Aug 24, 2013 at 2:29
  • $\begingroup$ @rghthndsd $8$ was enough to put the problem far beyond any feasible numeric computations. Of course, it wasn't essential. Anyway, I reformulated the problem. $\endgroup$
    – Լ.Ƭ.
    Aug 24, 2013 at 3:45

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