# 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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### Peter Walker's $C^{\infty}$ conjectured nowhere analytic slog

Consider the function $h(x)$ which is conjectured to be $c^\infty$ nowhere analytic, and can be used to generate what we call the base change slog, which is the inverse of the basechange sexp. This ...
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### Is $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}$ rational, algebraic irrational, or transcendental?

Let x be the expression. Assume x is algebraic irrational. By Gelfond-Schneider Theorem, $x^x$, which is also $x$, is transcendental, a contradiction. But I have no idea how to do the rest.
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### Prove if $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ than $z^{z^{z^{…}}}$ does not converge

Let $z \in \mathbb{C}$ and let $W$ be the Lambert W function. I am trying to prove that: If $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ for all $n \in \mathbb{N}$ (ie, $t$ is not a root of unity) ...
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### Infinite tetration of $-2.5$

Let $a_n$ be the sequence $z, z^z, z^{z^z} ...$ for $z \in \mathbb{C}$. This is sometimes called the iterated exponential with base $z$. I am investigating the above sequence for $z = -2.5$. After ...
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### Inverse operation of tetration and how it is computed?

If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$. What are the analogous inverse ...
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### What is this operator called?

If $x \cdot 2 = x + x$ and $x \cdot 3 = x + x + x$ and $x^2 = x \cdot x$ and $x^3 = x \cdot x \cdot x$ Is there an operator $\oplus$ such that: $x \oplus 2 = x^x$ and $x \oplus 3 = {x^{x^x}}$? ...
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### Solve $(x+a)^{1/x} = b$ for $x$

Solve $(x+a)^{1/x} = b$ , for $x$ where $a$ & $b$ are real constant. Do not use Lambert W-function in solution. Instead of using Lambert W-function, there are solution steps look like "...
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### The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
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### Equality of power towers : $a\uparrow\uparrow m=b\uparrow \uparrow n$

Suppose, $a,m,b,n$ are natural numbers greater than $1$. If we have $$a\uparrow\uparrow m=b\uparrow\uparrow n$$ can we conclude $a=b$ and $m=n$ ? $a\uparrow \uparrow m$ is a powertower of $m$ $a's$ ...
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### Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
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### What is a puiseux series and what is wolfram-alpha doing with this antiderivative?

I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the ...
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### 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 $n \geq 1$, ...
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### Last three digits of tetration

Find the last three digits of the number: $7^{7^{7^7...}}$ where there are 1001 sevens. I know how to do it for when there are 4 and 5 sevens. I get an answer of 343. But how do I find it for ...
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### Sum of the reciprocal of tetration?

Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth. What are f(2) and f(3), and could you please also explain how you reached these answers?
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### Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence

Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. ...
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### Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
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### How would you define non-integer tetration? [duplicate]

Tetration is defined for all $n\in \Bbb{N}$ by $${^1}a = a \\ {^{n+1}}a = a^{\left({^n}a\right)}$$ Thus ${^3}a$ means $a^{a^a}$. Here $a$ could be any real (or indeed even complex) value, but only ...