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I have a curious question. Let $x∈\mathbb{R}^+$ such that $x^x=2$. I am aware that the Gelfond–Schneider theorem implies that $x$ cannot be algebraic. However, is it still possible that $x$ can be expressed in terms of elementary functions applied to rationals? I saw this post but that is concerning the whole Lambert-W function and not a single point. Is anything known about this $x$? If not, is anything known about the cardinality of $\{ x : x∈E ∧ x^x=c ∧ c∈\mathbb{Q}^+ ∖ \{ k^k : k∈\mathbb{Z}^+ \} \}$, where $E$ is defined as the set of all reals that can be expressed using elementary functions applied to rationals?

By "elementary function applied to rationals" I mean any of the following:

  • Rational functions with rational coefficients
  • Exponential function
  • The inverse of any of the above
  • The composition of any of the above
  • Constants $0,1,i$

Since some people want a precise definition of "inverse", here is one: For any meromorphic function $f$, let $f^{-1}(z)$ be defined as $w = re^{it}$ where $r∈\mathbb{R}_{≥0}$ and $0≤t<2π$ iff $f(w) = z$ and $r$ is minimum and then $t$ is minimum. If no such $w$ exists then $f^{-1}(z)$ is undefined.

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    $\begingroup$ As usual for questions of this sort, you need to define what an elementary function is. The most obvious definition allows constant functions, but then all reals are so expressible! Chow's "What is a closed-form number?" (MSN) provides one possible way of formulating the question. $\endgroup$
    – LSpice
    Commented Mar 13, 2020 at 3:46
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    $\begingroup$ @CarloBeenakker OP linked this exact thread in the question. As they remark, the linked post is about the function, but this question is (essentially) about specific values of Lambert W. Even nonelementary functions can take elementary values. $\endgroup$
    – Wojowu
    Commented Mar 13, 2020 at 11:00
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    $\begingroup$ Won't every $x\in E$ belong to the set you query about? Since if $x$ is elementary, so is $x^x$. The less trivial question is for $x^x\in E$ but $x\not\in E$; presumably "most" (in some sense) $x$ with $c=x^x\in E$ will not belong to $E$, like the example with $c=2$ (which I would bet many of my marbles on not being elementary). $\endgroup$
    – Wojowu
    Commented Mar 13, 2020 at 11:21
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    $\begingroup$ The question is good but difficult. An easier qiestion would be wheher the number $x$ can be computed in linear time (that is, there is a Turing machine computing an $n$th decimal (or binary) approximation of $x$ in linear time. $\endgroup$
    – user6976
    Commented Mar 13, 2020 at 16:10
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    $\begingroup$ @MarkSapir I don’t think that’s any easier, as we lack tools to separate linear time from slightly super-linear time. (Note that the number is computable in time $O(M(n)\log n)$, see e.g. Wikipedia; that’s $O(n(\log n)^2)$ using the best known multiplication algorithm.) Also, I’m not sure how is it relevant, as the vast majority of constants defined by elementary functions are also likely not computable in linear time. $\endgroup$
    – Emil Jeřábek
    Commented Mar 13, 2020 at 17:20

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This is really an extended comment.

As Dan Richardson explains in his paper The elementary constant problem, there are different classes of numbers that you might be interested in. Your number certainly belongs to what is now usually called the class of elementary numbers, because it is a solution to the system of equations $$\eqalign{e^z &= 2 \cr xy&=z \cr e^y&= x\cr}$$ Now one could ask whether your number is a Liouvillian number, meaning that it's obtained by a finite sequence of algebraic, exponential, or logarithmic extensions of the rationals. But this allows for a more generous definition of "inverse" than you seem to want. As LSpice noted in a comment, you could also ask whether your number is a closed-form number in the sense of my paper, but this may be a more restrictive class than you want.

In any case, I think that the answer to your question is likely to be unknown since these types of questions tend to be very difficult.

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  • $\begingroup$ Thank you for your answer! Yes, this was just a question out of curiosity, and I just want to know if any result of this sort is known, whether with more or less restrictive classes of 'nice' numbers. Ordinarily, I expect such questions to be beyond known mathematics, but I have seen surprising results (to me) before. $\endgroup$
    – user21820
    Commented Mar 14, 2020 at 4:23

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