# Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

317 questions
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### Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not ...
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### Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
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### Explain $x^{x^{x^{{\cdots}}}} = \,\,3$

$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$ This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...
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### Calculating 7^7^7^7^7^7^7 mod 100

What is $$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$ I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
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### How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
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### Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
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### Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
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### How to solve $x^x = y^y \mod p$?

Let $p>5$ be a prime. How to solve $x^x = y^y \mod p$? How many solutions are there for a given $p$ such that $x,y < p$? I know the discrete logarithm and the theory of quadratic residues, but ...
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### Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. $y=\sqrt{6+\sqrt{6+\... 1answer 625 views ### What does$ ^3 3$mean? I was taking a test and this was a question. It does not mean$3^3$, nor is it a typo. The superscript was before the$3$and I have no idea what it means. I tried researching but couldn't find an ... 1answer 145 views ### Function$f$s.t.$\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$The questions are: 1) Does there exists some function$f$s.t.$\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$and$\lim_{x\to\infty}f(x)=\infty$? 2) Is$\big(\sum_{k=n}^{2^n}a_k\big)_n\to0$is ... 3answers 549 views ### What is the value of$^{\frac{1}{10}}e$? By$^nx$, I mean$x$tetrated to$n$. So, basically, I'm looking for the solution of the equation $$\large x^{x^{x^{x^{x^{x^{x^{x{^{x^x}}}}}}}}}=e$$. Is there some way to find the approximate value by ... 3answers 820 views ### Convergence properties of$z^{z^{z^{…}}}$and is it “chaotic”$\DeclareMathOperator{\Arg}{Arg}$Let$z \in \mathbb{C}.$Let$b = W(-\ln z)$where$W$is the Lambert W Function. Define the sequence$a_n$by$a_0 = z$and$a_{n+1} = {a_0}^{a_n}$for$n \geq 1$, ... 3answers 360 views ### Solutions of$a^{a^x}=x$for fixed$a>0$I am interested in the equation$a^{a^x}=x$for some fixed$a>0$. Is there some way to rearrange for$x$or solve otherwise? What about the nature of the solutions? For which fixed$a>0$are ... 1answer 161 views ### Does$\lim_{n\rightarrow \infty} \left(r+\frac{1}{n^2}\right)\uparrow \uparrow n=e$hold? Does $$\lim_{n\rightarrow \infty}\left (r+\frac{1}{n^2}\right)\uparrow \uparrow n=e$$ hold ?$r$is the number$e^{e^{-1}}$, the largest real number for which the infinite power tower$r\uparrow r\...
To solve equations involving power towers (infinite tetration) we usually do something like this: $$x^{x^{x^{x^{\dots}}}} =k$$ $$x^{(x^{x^{x^{\dots}}})} =k$$ $$x^k=k$$ $$x=\sqrt[k]k$$ But what if ...