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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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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 ...
Manuel's user avatar
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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 ...
mpear617's user avatar
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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.
tmlen's user avatar
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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 ...
SebbyIsSwag's user avatar
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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 ...
Matthias Liszt's user avatar
4 votes
2 answers
218 views

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 ...
mathy_mathema's user avatar
1 vote
1 answer
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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 ...
Marco Ripà's user avatar
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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, ...
alati ahmad's user avatar
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Tetration uniqueness by $ A = \inf \sum_n a_n^2 $?

Let $x > -2$ and $f(0) = 1,f(x+1) = \exp(f(x))$ And $f$ is a taylor series : $$f(x) = \sum_n a_n x^n$$ where the $a_n$ are all real and all nonzero and $f$ has radius $2$. (Notice $f(-1) = 0, f(-2) ...
mick's user avatar
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Exponential Fibonacci and its recurrence-relation to ϕ. [closed]

1,1,2,3,5,8,... = Additive Fibonacci: a(n-1) * phi is asymptotic to a(n) 2,2,4,8,32,256,... = Multiplicative Fibonacci a(n)=a(n-1)*a(n-2): a(n-1) ^ phi is asymptotic to a(n) 2,2,4,16,65536,1.158*10^77....
Peter Woodward's user avatar
2 votes
1 answer
166 views

Might results which show the same result for tetration as for exponentiation be of any use (like in the range from 2 to e^(1/e))?

I experimented with this and found 9 numbers which have the same height and exponent and show nearly the same result for tetration and exponentiation. Might this be of any use or worth looking at it ...
Matthias Liszt's user avatar
1 vote
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Tetration and $f(x) = \exp(\int_1^x \ln(f(t)) dt)$

Let $g(x)$ satisfy $g(1) = 1 , g'(1) = 1 , g(1+x) = \exp(g(x))$ Now it is clear that $g'(5) = g(5)g(4)g(3)g(2)$ This invites to think of the function $f(x)$ which is defined similarly and might or ...
mick's user avatar
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Which hyperoperations produce a "prefix-complete" sequence?

Definition ("prefix-complete"): A sequence of positive integers $(a_n)_{n=1,2,3,\dots}$ will be called prefix-complete in base $b$ iff, for any positive integer $p$, there is some $a_n$ ...
r.e.s.'s user avatar
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Sources on how tetration is defined in set theory and on infinite power towers

I was able to find the set theory definitions of addition and multiplication, but not of tetration. I wondered if somebody could define tetration in terms of set theory, and hopefully provide some (...
string_knot's user avatar
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Tetration Power Series

While reading through the Citizendium article on tetration, the first hyper-operation above exponentiation, I came across a power series approximate of tetration. The article said that it got the ...
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Definition of tetration [duplicate]

We all know that many years ago we invented powers. e.g. $3^4$ meant how many times we multiply 3. i.e. $3^4=3\cdot 3\cdot 3\cdot 3$. But then, people started asking questions like what is the ...
Chess player's user avatar
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Could we define complex hyper operators?

I made a similar question in the past. Let's define the function $H_n(a,b)$ to be the $n$-th hyperoperation which imputs are $a$ and $b$. $H_1(a,b)= a+b~~$ Addition $H_2(a,b)= ab~~$ Multiplication $...
Pierre Carlier's user avatar
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217 views

What are all the tetration extension methods?

Tetration is the next step in our regular operations. Addition, multiplication, exponentiation, tetration. It is constructed by repetitive exponentiations. "$a$ tetration $b$" is written $^{...
Pierre Carlier's user avatar
2 votes
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40 views

Do operators exist in between the common ones? [closed]

I might be in over my head asking this question as I am only a rising sophomore in high school having just finished algebra 2, so I probably won't understand any complicated math concepts, but I'll ...
Robocittykat's user avatar
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Solve $^y{x} =$ $^x{y}$ over the real numbers

Let $x, y \in \mathbb{R}^{+}$ be such that $x \neq y$ and assume $n \in \mathbb{N}-\{0\}$. Now, referring to the well-known hyperoperation sequence $x[n]y$, we have that $x[1]y=x+y$ and we know that ...
Marco Ripà's user avatar
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2 votes
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Is there a set-theoretic construction of tetration and even higher-order Ackermann functions?

Cardinal addition, multiplication, and exponentiation have set-theoretic constructions, namely disjoint union, cartesian product, and the set of all functions from $S$ to $T$, respectively. Are there ...
user107952's user avatar
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216 views

Modular tetration (power tower) for non-coprime numbers case

I'm writing algorithm for calculating $a^{a^{...^{a}}}$ mod $m$. According to Euler's theorem, $a^k = a^{k\mod{\phi(m)}}$ mod $m$, if $a$ and $m$ are relatively primes. If $m = \prod_{i=0}^n p_i^{\...
Vitaliy Volovyk's user avatar
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80 views

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 ...
rubiko's user avatar
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201 views

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 $c \in \mathbb{Z}^+$ ...
Marco Ripà's user avatar
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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 ...
mick's user avatar
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2 votes
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99 views

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 ...
Тyma Gaidash's user avatar
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1 answer
106 views

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?
Phenomniverse's user avatar
2 votes
1 answer
98 views

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 ...
Graviton's user avatar
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2 votes
1 answer
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Does the distributive property apply to all hyperoperations higher than multiplication?

Suppose we have a function defined like so; $$f_n( x, x ) = f_{n+1}( x, 2 )$$ $$f_n( x, x, x ) = f_{n+1}( x, 3 )$$ $$f_n( x, x, x, x,\dots) = f_{n+1}( x, \text{number of} x\text{'s} )$$ $f_1( x, y ) = ...
Caleb Thoburn's user avatar
1 vote
0 answers
289 views

Do we need a new number system for inverse tetration?

I know that the simplest number system is the natural numbers ($0, 1, 2, …$). And while we can easily define addition and multiplication for naturals, subtraction doesn’t work. Because what would $2-3$...
Zachary Sakowitz's user avatar
2 votes
1 answer
99 views

tetration primitive root $q \mod p$

Consider primitive roots $q \mod p$ where $q$ is a prime and $p$ is an odd prime $> 5$. I am looking for such pairs $q,p$ such that every residue $a_i \mod p$ is of the form $$a_i = q^{(v_i)} \mod ...
mick's user avatar
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1 vote
1 answer
384 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. Equation A $$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $$ or variations of it like the weaker Equation B $$f(f(f(f(z)))) = z , ...
mick's user avatar
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7 votes
1 answer
197 views

Evaluate: $\sum\limits_{n\ge1}\sum\limits_{m\ge0}\sum\limits_{k=0}^{n-1}\frac{y^nm^k(-n)^m\delta_{k+m-n+1}}{(k+m-n+1)!\Gamma(n-k)k!n}$

Context: The cube super root ssrt$_3(x)$ series expansion yielded part of it as: $$\sum_{n=1}^\infty\frac{y^n}{n!}\sum_{m=0}^\infty\frac{(-n)^m}{m!}\sum_{k=0}^{n-1}\binom{n-1}k\left.\frac{d^kt^m}{dt^k}...
Тyma Gaidash's user avatar
4 votes
1 answer
50 views

Proving that a family of complicated functions is less than one particularly simple function

I claim that $$f_b(x) = 1 - \left(1 - 1/x^b\right)^{x^{b-1}} \le 2/x = g(x)$$ is true for every positive integer $b$ and for all $x \ge 1$. The claim appears to be true (even for non-integer $b$) when ...
Utkan Gezer's user avatar
2 votes
2 answers
289 views

How to use $\uparrow$ to define an explicit bijective mapping $f:\varepsilon_{1}\rightarrow\mathbb{N}$?

The map $f:\varepsilon_{1}\rightarrow\mathbb{N}$ which I am trying to define has to send $\varepsilon_{0}$ to some natural number. Since $\varepsilon_{0}=\omega\uparrow^{2}\omega$, a potential ...
John's user avatar
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0 votes
1 answer
99 views

Sign of $n$ th derivative of $f(x)$?

Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$ More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function. see : https://en.wikipedia.org/...
mick's user avatar
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1 vote
2 answers
156 views

Infinite tetration of $(-1)$?

For all positive integers $n$, $^n(-1) = -1$, thus I thought ${^{\infty}}(-1)$ could be $\displaystyle \lim_{n \to \infty} {^n}(-1) = -1$. But $\dfrac{W(-\ln z)}{-\ln z}$, analytic continuation of ...
user404273's user avatar
2 votes
0 answers
58 views

$\int_{0}^{1}x^{x^{\alpha}} dx$ for negative $\alpha$

I'm trying to come up with a formula for $$\int_{0}^{1}x^{x^{\alpha}} dx,$$ basically the Sophomore's Dream. $$\sum_{k=0}^{\infty}\frac{(-1)^k}{(\alpha k+1)^{k+1}}$$ However it only works for positive ...
Noa Arvidsson's user avatar
3 votes
0 answers
244 views

$\int f^{-1}(\frac{1}{x}) dx $, where $f^{-1}$ is inverse function. [closed]

I had to changed orginal question because I was too edgy with my different questionsand I can't ask another one It can by showed that $\int f^{-1}(x) dx=xf^{-1}(x)-F \circ f^{-1}(x)+C$ Is there exists ...
Wreior's user avatar
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6 votes
1 answer
162 views

Branch of mathematics that deals with repeated operations

One interesting trait of subtraction is that it can introduce us to negative numbers using just positive whole numbers. For instance, $1-3=-2$. Division, similarly, can introduce us to a new set of ...
Frankie S's user avatar
8 votes
0 answers
247 views

Sharp bounds for power towers $x^x,x^{x^{x^x}},x^{x^{x^{x^{x^x}}}},\cdots$

I am looking for a reference to results on sharp upper and lower bounds for the $2n$th power towers $$x^x,x^{x^{x^x}},x^{x^{x^{x^{x^x}}}},x^{x^{x^{x^{x^{x^{x^x}}}}}},\cdots$$ over the intervals $x\in[...
TheSimpliFire's user avatar
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95 views

Is there such thing as an exponential/logarithmic inverse?

Okay, let me explain. First, there's the additive inverse, or the negative form of a number. i.e. the additive inverse of $2$ is $-2$. Then, there's the multiplicative inverse, or the reciprocal of a ...
Dawson Piercey's user avatar
0 votes
2 answers
72 views

Equivalence of Roots on the Exponential Level - $a^a$ ($n$ $times$) $ = b,$ $find$ $a$

My question is how would the concept of roots, on the multiplication level, be expanded to the exponential level? For example, the equivalence of roots on the addition level is division as to find $a$ ...
Nov's user avatar
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7 votes
1 answer
285 views

Is it possible to express logarithm with tetration?

Subtraction and division can be expressed with multiplication and exponentiation, as follows: a - b = a + (b * -1) a / b = a * (b ^ -1) My question is: does this ...
MaiaVictor's user avatar
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2 votes
1 answer
162 views

A really really weird new(to my knowledge) kind of differential equation.

the equation $\dfrac {d^xf}{(dx)^x)} = f(x)$ where $ \dfrac{d^xf}{(dx)^x}$ means we are taking the xth derivative of f(x)(using fractional calculus, assuming the Riemann–Liouville fractional ...
Colonizor48's user avatar
0 votes
0 answers
100 views

Knuth down-arrow notation

I was reading this article on MathWorld: https://mathworld.wolfram.com/DownArrowNotation.html, and I decided to check the statement $\ln^{*}n$ is the number of times the natural logarithm must be ...
Yuri Kotsar's user avatar
0 votes
0 answers
81 views

Integral with Lambert $W$ function [duplicate]

I'm trying to calculate the integral of the infinite tetration of $x$ where it's defined, $$\displaystyle\int_{e^{-e}}^{e^{1/e}}x^{x^{x^{x^{x^{...}}}}}dx$$ which simplifies to $\displaystyle\int_{e^{-...
Noa Arvidsson's user avatar
11 votes
2 answers
374 views

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 ...
Colonizor48's user avatar
3 votes
1 answer
215 views

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 ...
Tsar Asterov XVII's user avatar
2 votes
0 answers
659 views

How to approximate the result of a tetration

I'm trying to calculate the result of some tetrations of 2 but my program can't get to 2↑↑6, it just takes to long. So i wanted to ask if there's a way to approximate the result of a tetration, even a ...
Creator565's user avatar

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