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# 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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### Has something like “Knuth's Up-Arrow Factorial Notation” ever been used? If so, what practical uses does it have?

I was studying Knuth's up-arrow notation and I was wondering if ever something like "Knuth's up-arrow factorial notation" has ever been used. Now I know this probably isn't a recognizable term, ...
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### For which $a\in\Bbb R^+$ is $\{1,a, a^a, a^{a^a},…\}$ linearly independent over $\Bbb Z$?

Can I choose a positive real number $a\in\Bbb R^+$ so that $1,a,a^a,a^{a^a},...$ are independent in the sense that no combination of integer coefficients will add up these numbers to zero? More ...
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### How many ways can a sequence of $1$s be partitioned into pairs or singles?

How many distinct ways can a sequence of $n$ $1$s be partitioned into pairs or singles, in which $\{1,1\}=\{2\}$ is considered a pair and $\{1\}$ is considered a single? For example $\{1,1,1,1\}$ can ...
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### Infinite Power Tower

I've been having fun with the problem of finding the values of $n$ for which the infinite power tower $$\sqrt{2}^{\sqrt{2}^{...^{\sqrt{2}^n}}}$$ Has a finite value. My final answer was that it ...
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### How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\...
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### How do you calculate $2^{2^{2^{2^{2}}}}$?

From information I have gathered online, this should be equivalent to $2^{16}$ but when I punch the numbers into this large number calculator, the number comes out to be over a thousand digits. Is the ...
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### Derivative of super square root [closed]

What is the derivative of $y=^{1/2}x$? I tried finding the derivative of $x^{x}$ and then finding the inverse of that, but that didn't work.
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### Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations?

Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the ...
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### Can any number of factorials overtake the tetration function ${^x}10$?

Tetration is repeated exponentiation evaluated from right to left. The value of both the factorial and tetration function at $x=0$ is defined to be 1. So,both functions start at $x=0$ (when the values ...
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### How can I define a function that raises a number to the power of itself a given number of times?

We can use the following to add the number $2$ to itself $5$ times. $$f(n,k) = \sum_{x=1}^k n = n\cdot k$$ $$2 + 2 + 2 + 2 + 2 = f(2,5) = \sum_{x=1}^5 2 = 2\cdot 5 = 10$$ We can use a similar ...
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### Finding the value of n, so that it is bigger than M?

We introduce some notation for writing really big (but finite) numbers. A googol, denoted g, is defined by $g = 10^{100}$. A googolplex, denoted G, is defined by $G = 10^g$. A MathPatharoo, denoted M, ...
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### When does $x^{x^{x^{…^x}}}$ diverge but $x^{x^{x^{…^c}}}$ converge?

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $c\ne x^c$ $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ to converge under ...
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### Find a sequence such that this tower of of exponent is convergent

Context We already know that if we take a sequence $(x_n)\in{\mathbb R_+^*}^{\mathbb N}$ such that $$x_n=O\left(\frac 1{n^2}\right)$$ then $$\sum_{n=0}^\infty x_n <+\infty.$$ We also now that ...
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### Asymptotic to $f( 2 f^{[-1]}(x) )$? [closed]

Let $f(x)$ be the half-iterate of $2 sinh(x).$ Im looking for an asymptotic to $f( 2 f^{[-1]}(x))$ for Large $x>0$. $^{[*]}$ means iteration here thus $^{[-1]}$ means functional inverse. For the ...
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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 $$x=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}.$$ Assume $x$ is algebraic irrational. By the Gelfond-Schneider Theorem, $x^x$, which is also $x$, is transcendental, a contradiction. But I ...
### 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) ...