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Let's solve the (real) equation $$x^{x^{x^{\dots}}}=2\qquad (x>0).$$ If we call $E(x):=x^{x^{x^{\dots}}}$, it holds that $x^{E(x)}=E(x)$, but our equation says that $E(x)=2$, so it holds that $x^2=2$, whence $x=\sqrt2$.

Noew, lets solve $$x^{x^{x^{\dots}}}=4\qquad (x>0).$$ A similar arguments, leads us to to solve $x^4=4$ whose solutions are $x=\pm\sqrt2,\pm i\sqrt2$, so the only positive solution is, again $x=\sqrt2$.

But this yields to the contradiction that $2=4$.

So, my question is... what's wrong here?

My intutition says that is has something to do with the complex solutions of $x^4=4$, but I'm not really sure.

Als0 it may something to do with the convergence of the next sequence: $$S_1(x)=x>0,\qquad S_{n+1}(x):=x^{S_n(x)}$$ but I'm not able to study its convergence.

Tito Eliatron
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    I think this exact question is discussed in this Power Tower video by 3Blue1Brown. – angryavian Jan 11 '21 at 18:18
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    This post addresses the same issue (but with different numbers) – Ben Grossmann Jan 11 '21 at 18:18
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    The domain of $x^{x^{x^\cdots}}$ is only $(e^{-e},e^{\frac{1}{e}})$. The lower bound can be extended to $0$ for even or odd subsequences of the infinite tetration limit, but the upper bound is firm. – Ninad Munshi Jan 11 '21 at 18:22
  • Thank's a lot. Very interesting video @angryavian – Tito Eliatron Jan 11 '21 at 18:30
  • @NinadMunshi I think it is worth noting that $\sqrt[x]{x}=e^{\frac1e}$ when the limit of the power tower is $x=e$ while $\sqrt[x]{x}=e^{-e}$ when the limit of the power tower is $x=\frac1e$ – Henry Jan 11 '21 at 18:33
  • To continue what @Ninad Munshi said, the issue is one of convergence (existence of value). A possible analogy is $x = 1 + 2 + 4 + 8 + \cdots.$ Multiplying both sides by $2$ gives $2x = 2 + 4 + 8 + \cdots,$ and since $2 + 4 + 8 + \cdots = (1 + 2 + 4 + 8 + \cdots) - 1,$ we get $2x = x - 1,$ or $x = -1.$ The problem? $1 + 2 + 4 + 8 + \cdots$ doesn't represent a real number (in the sense that continuing the sum further and further gets you closer and closer to a single real number), and thus manipulating it as if it were a real number is basically a case of "garbage in $\rightarrow$ garbage out". – Dave L. Renfro Jan 11 '21 at 18:36
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    but I'm not able to study its convergence --- For a lot of references on this topic, see Ioannis Galidakis's list of references. (FYI, I snail-mailed him photocopies of these papers over a period of several years back in the early 2000's and suggested it would be a good idea to post a bibliographic list in his web pages.) – Dave L. Renfro Jan 11 '21 at 18:45

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