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Im trying to solve for the exact solution of

$$ x^{x^x} = 17 $$

I understand that the previous solution, $x^x=17$, does not have an exact closed form solution and requires use of the Lambert W function, however i have not found an answer (from wolfram included) that seems to provide a non approximated answer even without said constraints. When approximating, however, the final answer appears to be close to $2.009$.

Does anyone know of a closed-form solution to this problem, be it even if it involves use of non-elementary functions?

IV_
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Jacob
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  • It can be shown that the function $F(x):=x^{x^x}$ is strictly increasing on $[1,\infty)$ and that $F(1)=1$. Then, there exists a unique function $F^{-1}:[1,\infty)\mapsto[1,\infty)$ such that $F^{-1}\circ F=\mathrm{Id}$. The exact solution is then $F^{-1}(17)$ and $F^{-1}$ is non-elementary. – KBS May 02 '22 at 13:33
  • Just for the fun $x=2+\frac{1}{100} \left(\frac{1}{3}+\cos (1)\right)^{\frac{1}{4} \left(1+\sqrt{5}\right)}$ gives $16.999999822$ – Claude Leibovici May 02 '22 at 14:41
  • Why bother about a possible exact solution , if we can easily determine the solution with arbitary precision ? Even for theoretical purposes, such an expression would be neither useful nor enlightning. Interesting, but probably out of reach would be a proof that the solution is transcendental (already an irrationality proof would be interesting). – Peter May 02 '22 at 16:53

2 Answers2

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If you want a definition of the exact value, consider that it is the limit, when $n\to \infty$, of the first iterate of a Newton-type method of order $n$.

Thsi will give for example, with $L=\log(2)$ $$x_{(2)}=\frac{65+128 L+128 L^2}{32+64 L+64 L^2}$$ $$x_{(3)}=\frac{274+1084 L+2136 L^2+2120 L^3+1056 L^4}{135+538 L+1064 L^2+1060 L^3+528 L^4}$$ $$x_{(4)}=\frac{ 27735+164754 L+487008 L^2+855140 L^3+951704 L^4+630240 L^5+209024 L^6 } {13665+81570 L+241908 L^2+425980 L^3+475060 L^4+315120 L^5+104512 L^6 } $$

The first decimal values $$\left( \begin{array}{cc} n & x_{(n)} \\ 2 & \color{red}{2.00}93361604829833 \\ 3 & \color{red}{2.00896}17010469369 \\ 4 & \color{red}{2.0089646}624948464 \\ 5 & \color{red}{2.0089646909}446277 \\ 6 & \color{red}{2.0089646909}923880 \\ 7 & \color{red}{2.008964690988}6512 \\ 8 & \color{red}{2.00896469098858}31 \\ \cdots & \cdots \\ \infty & \color{red}{2.0089646909885827} \end{array} \right)$$

Claude Leibovici
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You ask for an exact solution. That means solutions in closed form.

Your equation is a case of the more general equations asked for in Solving $x^{x^x}=c$ with Lambert's W function.

The function $f$ with $f(x)=x^{x^x}$ doesn't have partial inverses over non-disrete domains in terms of elementary functions and/or Lambert W. Therefore, your equation $x^{x^x}=17$ cannot be rearranged for $x$ by applying only Lambert W and/or elementary operations (means elementary functions) you can read from the equation.
But it is an open question if the equation has solutions in the elementary numbers.

Equations of your kind can be solved by generalized hyper-Lambert functions (see the references below).
$\ $

Fantini, J.; Kloepfer, G.: Wexzal/The coupled Exponent, 1998

Helms, G.: Wexzal"/"coupled exponent" , "Superroot" and a generalized Lambert-W

Galidakis, I. N.: On solving the p-th complex auxiliary equation $f^{(p)}(z)=z$. Complex Variables 50 (2005) (13) 977-997

Galidakis, I. N.: On some applications of the generalized hyper-Lambert functions. Complex Variables and Elliptic Equations 52 (2007) (12) 1101-1119

Galidakis, I.; Weisstein, E. W.: Power Tower. Wolfram MathWorld

IV_
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