# 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.

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### Growth of factorial functions and tetration

Good morning everyone, I have a doubt about the growth rate of the following two functions: $$\operatorname f(x)=x!$$ and $$\operatorname g(x)=^x n,\quad\mbox{where}\ \quad n \in \mathbb{R^+}$$ I ...
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### Is $x^x = (x-1)^{x+1}$?

Background: I was trying to estimate the size of $21^{21}$ for some problem and decided to use $20^{22}$ as hopefully a rough approximate ($20^{22} = 2^{22} \cdot 10^{22} \approx 10^{28}$). But then I ...
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### Solution of $x^x = (x-1)^{x+1}$ [closed]

Is there any way to solve this equation algebraically and give an exact form of the solution: $x^x = (x-1)^{x+1}$? WolframAlpha only finds the approximate solution 4.14.
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### How to interlopate the pickover factorial?

Question I want to be able to interlopate the pickover factorial, defined as: $$n = ^{n!}n!$$ where $^xx$ represents tetration. I want to be able to interlopate this function. Context The reason I ...
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### Could an equation like ${^{h}b} = b^h +$c together with polynomial interpolation be used to calculate tetration? Are there flaws in it?

As it is quite easy to calculate integer solutions for $\;^hb = b^h$ the question arose on how to find a solution for real heights. One idea was of predicting the height by finding a formula ...
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### Finding the convergent interval of $z^{{2z}^{{3z}\ldots^{{nz}\ldots}}}$

I was trying to find an exact solution to $x=a^x$, and naturally derived a solution defined by infinite tetrations of $a$. I first defined a recursion equivalent to the definition of tetrations and ...
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### Contradiction occurred trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ [closed]

Today I was trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ (i.e., $\sqrt{2}$ plus the real part of the tenth tetration of the base $-\sqrt{2}$) up ...
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### Infinite exponentiation [duplicate]

It started as an exercise for my students. Calculate $i^i$, then $i^{i^{i}}$ and make a conjecture if we follow that pattern. If we define $u_n=i$ and $$u_{n+1}=i^{z_n}=e^{i\frac{\pi}{2}z_n}$$ Then, ...
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### Infinite power tower for $x > e^{1/e}$

For $x>0$, let $\tau_0 = 1$ and $\tau_{n+1} = x^{\tau_n}$. The infinite power tower of $x$ is then $\tau = \tau(x) = \lim_{n\to \infty} \tau_n$. It is well known that $\tau$ exists and is finite ...
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### Are there infinitely many $c$-th perfect powers having a constant congruence speed of $c$?

Let $a$ be a positive integer of the form $20 \cdot n + 5$ (i.e., $a : a \equiv 5 \pmod {20}$, $n \in \mathbb{N}_0$). I wish to prove (or disporove) the following statement. Let $c \in \mathbb{Z}^+$ ...
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### Iterating $\tanh(A x)$ and $\lim_{x \to +\infty} \tanh^{[r]}(A x) = C_r$?

Everybody who ever studied special relativity or hyperbolic trig knows this function $$\tanh(Ax)$$ for real $0 < A < 1$ $\tanh(A x)$ has a nice attracting fixpoint at $0$ and it is asymptotic to ...
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### Evaluating $\frac{d^n}{dt^n}e^{a(t-e^t)}$ as a single series to extend region of convergence for super root function

$\def\srt{\operatorname{srt}}$Introduction: There is a multiple series expansion for the super root $\srt_n(z)$ valid near $0.7<|z|<1.4$. However, for around $0<|z|<1.3$, there is this ...
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### How to find what number raised to its own power is equal to a given number? [duplicate]

If $x^x = 3$, how can I find x? I know I can rearrange this to $\log_x(3) = x$, and that some calculators can solve this, but how would you do this manually?
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### Closed form for $H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-k, j)j^k$ where $H(1,k)=\frac{1}{k!}$

Let $H(n,k)$ be defined such that $$H(1,k)=\frac{1}{k!}\text{, and }H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-1,j)j^k$$ As pointed out in the comments, I should mention that we must define $0^0=1$ as ...
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### Can the second integral of $x^x$ be expressed in terms of the first integral and standard mathatical functions?

Note: by elementary I also mean functions like $\operatorname{Li}(x)$ and $\operatorname{Erfi}(x)$. Edit: This is not a duplicate. I am not asking if the integral of $x^x$ is elementary. Im asking if ...
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### How to compute tetration of values where the value $k$ is a negative integer? [duplicate]
I would like to know about how to exactly do calculation with tetration, especially when the value $k$ is a negative value in: $a ↑↑ k$ I am aware of the process of tetration, which is repeated ...