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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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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How can the domains of all variables of this hyperoperation function be extended to the entire complex plane?
When generalizing addition, multiplication, and exponentiation, there exists a certain hierarchy upon which these operators can be placed: the hyperoperator hierarchy. Starting with succession, each ...
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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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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 ) = ...
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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 $2-3$...
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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
$$a_i = q^{(v_i)} \mod ...
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$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?
While talking about tetration with my friend the following idea (re)occured.
Equation A
$$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 , ...
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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:
$$\sum_{n=1}^\infty\frac{y^n}{n!}\sum_{m=0}^\infty\frac{(-n)^m}{m!}\sum_{k=0}^{n-1}\binom{n-1}k\left.\frac{d^kt^m}{dt^k}...
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Proving that a family of complicated functions is less than one particularly simple function
I claim that
$$f_b(x) = 1 - \left(1 - 1/x^b\right)^{x^{b-1}} \le 2/x = g(x)$$
is true for every positive integer $b$ and for all $x \ge 1$. The claim appears to be true (even for non-integer $b$) when ...
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How to use $\uparrow$ to define an explicit bijective mapping $f:\varepsilon_{1}\rightarrow\mathbb{N}$?
The map $f:\varepsilon_{1}\rightarrow\mathbb{N}$ which I am trying to define has to send $\varepsilon_{0}$ to some natural number. Since $\varepsilon_{0}=\omega\uparrow^{2}\omega$, a potential ...
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Sequence of the most significant digits of the integer tetration ${^{b}a}$: is it (eventually) periodic?
Let $F(a,b) := F({^{b}a})$ indicate the first figure (i.e., the most significant digit) of the tetration ${^{b}a} : a,b \in \mathbb{Z}^+$.
As an example, we have that $a=2 \Rightarrow F(a,b)=2,4,1,6,2,...
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Sign of $n$ th derivative of $f(x)$?
Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$
More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function.
see : https://en.wikipedia.org/...
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Infinite tetration of $(-1)$?
For all positive integers $n$, $^n(-1) = -1$, thus I thought ${^{\infty}}(-1)$ could be $\displaystyle \lim_{n \to \infty} {^n}(-1) = -1$.
But $\dfrac{W(-\ln z)}{-\ln z}$, analytic continuation of ...
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$\int_{0}^{1}x^{x^{\alpha}} dx$ for negative $\alpha$
I'm trying to come up with a formula for $$\int_{0}^{1}x^{x^{\alpha}} dx,$$ basically the Sophomore's Dream.
$$\sum_{k=0}^{\infty}\frac{(-1)^k}{(\alpha k+1)^{k+1}}$$
However it only works for positive ...
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Tetration and its connection with Lambert W function
I was working on some stuff, where to my surprise Lambert W function appear. It shocked my where I realise that
$\displaystyle \frac {W(\ln(1/x))}{\ln (1/x)}=x^{\frac {W(\ln(1/x))}{\ln (1/x)}}=x^{x^{\...
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Branch of mathematics that deals with repeated operations
One interesting trait of subtraction is that it can introduce us to negative numbers using just positive whole numbers. For instance, $1-3=-2$. Division, similarly, can introduce us to a new set of ...
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Sharp bounds for power towers $x^x,x^{x^{x^x}},x^{x^{x^{x^{x^x}}}},\cdots$
I am looking for a reference to results on sharp upper and lower bounds for the $2n$th power towers $$x^x,x^{x^{x^x}},x^{x^{x^{x^{x^x}}}},x^{x^{x^{x^{x^{x^{x^x}}}}}},\cdots$$ over the intervals $x\in[...
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Is there such thing as an exponential/logarithmic inverse?
Okay, let me explain.
First, there's the additive inverse, or the negative form of a number.
i.e. the additive inverse of $2$ is $-2$.
Then, there's the multiplicative inverse, or the reciprocal of a ...
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Equivalence of Roots on the Exponential Level - $a^a$ ($n$ $times$) $ = b,$ $find$ $a$
My question is how would the concept of roots, on the multiplication level, be expanded to the exponential level?
For example, the equivalence of roots on the addition level is division as to find $a$ ...
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Is it possible to express logarithm with tetration?
Subtraction and division can be expressed with multiplication and exponentiation, as follows:
a - b = a + (b * -1)
a / b = a * (b ^ -1)
My question is: does this ...
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A really really weird new(to my knowledge) kind of differential equation.
the equation $\dfrac {d^xf}{(dx)^x)} = f(x)$ where $ \dfrac{d^xf}{(dx)^x}$ means we are taking the xth derivative of f(x)(using fractional calculus, assuming the Riemann–Liouville fractional ...
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Knuth down-arrow notation
I was reading this article on MathWorld: https://mathworld.wolfram.com/DownArrowNotation.html,
and I decided to check the statement
$\ln^{*}n$ is the number of times the natural logarithm must be
...
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Integral with Lambert $W$ function [duplicate]
I'm trying to calculate the integral of the infinite tetration of $x$ where it's defined,
$$\displaystyle\int_{e^{-e}}^{e^{1/e}}x^{x^{x^{x^{x^{...}}}}}dx$$
which simplifies to
$\displaystyle\int_{e^{-...
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Can the second integral of $x^x$ be expressed in terms of the first integral and standard mathatical functions?
Note: by elementary I also mean functions like $\operatorname{Li}(x)$ and $\operatorname{Erfi}(x)$.
Edit: This is not a duplicate. I am not asking if the integral of $x^x$ is elementary. Im asking if ...
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How to compute tetration of values where the value $k$ is a negative integer? [duplicate]
I would like to know about how to exactly do calculation with tetration, especially when the value $k$ is a negative value in:
$a ↑↑ k$
I am aware of the process of tetration, which is repeated ...
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How to approximate the result of a tetration
I'm trying to calculate the result of some tetrations of 2 but my program can't get to 2↑↑6, it just takes to long.
So i wanted to ask if there's a way to approximate the result of a tetration, even a ...
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Possible Tetration extension for a specific interval (part 2)
So this is part 2 of a previous post, so I advise you to watch the first one to better understand this post.
("x tetration r" = ${^r}x$).
Now, I'm gonna explain the method I use to extend ...
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Possible Tetration extension for a specific interval (part 1)
My friend and I have been developing an extension of tetration for non integer values.
We managed to get definitions of extensions for : ${^r}x$.
$x$>0.
$r$ not equal to any whole number below -1.
...
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Solving an infinite system of equations for the coefficients of a power series for $f(x+1)=\exp(f(x))$
Consider the sequence of formulae, $a_n$, such that $a_0=1$ and for all $n>0$, we have
$$a_n = \sum_{k=1}^nT(n,k)c_k a_{n-k},$$
where $$T(n,k)=\frac{(n-1)!k}{(n-k)!},$$
$c_0=0$, and $c_k$ for $k>...
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Solving $\frac{d^{n-1}}{da^{n-1}}(f^n(a)g’(a))=h(a,n)(n+u)^{vn},v<0$ from Lagrange reversion
The Lagrange Reversion theorem states
$$x=a+bf(x)\implies g(x)=g(a)+\sum_{n=1}^\infty\frac{b^n}{n!}\frac{d^{n-1}}{da^{n-1}}(f^n(a)g’(a))$$
Our goal is using the Lagrange reversion formula obtaining:
...
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Find f(32) given f(x) = f(x-1) + 2^(f(x-1)+1) and f(0) = 5 [closed]
I would like to find f(32) given:
f(x) = f(x-1) + 2^(f(x-1)+1)
f(0) = 5
I figured that this function grows tetrationally (if that is a word) but I don't really know
Where I got this equation from:
In ...
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Convergence value of the unique positive real root of ${^{m + 1}x} = {^{m}x} + 1$ as $m$ approaches infinity
As $m \in \mathbb{Z}^{+}$ gets bigger and bigger, the unique positive real root of the equation ${^{m + 1}x} = {^{m}x} + 1$, where ${^{m}x} := x^{x^{x^{\dots}}} \}m\text{-many} \; x\text{'s}$, gets ...
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Undefined value arising from a new property of tetration
In order to introduce the question, please let me provide the following explanation first.
In three published papers, I have shown the existence of a new property involving the integer tetration $$
\...
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Has There Been Any Study of the Quotient-root Derivative Definition $\lim_{\Delta x \to 0} \sqrt[\Delta x]{\frac{f(x + \Delta x)}{f(x)}}$?
Has there been a study of turning the difference-quotient seen in the common derivative
$$
\frac d {dx} f(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
$$
into a quotient-root
$$
...
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Does $^{\frac12}{a}=$ $\sqrt{a}_s$?
Just to be sure, does $^{\frac12}{a}=$ $\sqrt{a}_s$? I only ask because, although the Wikipedia page on tetration and other sources explain that the super-root is one of the inverse operations of ...
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Improving the result $\int_0^\infty\frac1{x^x}\, dx<2.$
Problem/Conjecture :
Let $a\geq 1$ be a real number then it seems we have :
$$\int_{0}^{\infty}x^{-x}dx<\int_{0}^{\infty}\left(h^{-\frac{1}{2}h^{\int_{0}^{1}y^{y^{e^{ah}}}dy}}\cdot h^{-\frac{1}{2}h^...
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Solve for $x$ in $^nx = i$
$\def\rddots#1{\cdot^{\cdot^{\cdot^{#1}}}}$
$$
\left.
\begin{array}{ll}
^nx := x^{x^{\rddots x}}
\end{array}
\right \}n\text{-many} \; x\text{'s} \tag{1}
$$
Using the definition in $(...
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What is $^{y(slog_xa)}x$?
What is $^{y(slog_xa)}x$ ?
In this website about tetration, it's shown that $^{y+slog_xa}x=$ $^{y}x^a$, as $^{1+slog_xa}x=x^a$.
So what is $^{y(slog_xa)}x$ ? Is it $^{y}a$ ?
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Negative Tetrations?
To start, I'll say that for this post I'll be using Rudy Rucker notation for tetration. That being $^2$x=$x^x%$, which means the number raised to the left means how many times one would exponentiate x....
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General formula for the derivative of a pentation?
Now, many people probably know of the first three hyperoperations, such as addition, multiplication, and exponentiation. However, many don't know of the fourth, tetration, and even less then know ...
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Can we characterize functions, like exponentials, the gamma function, and tetration, as solutions of an optimization problem?
This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to ...
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General Rule for Differentiation of Tetrations
I'll start at the beginning. Initially, this sort of began as just what is $\frac{d}{dx}$[$x^x$] the answer being $x^x$+ln(x)$x^x$. This wasn't difficult to achieve, just some chain rule and product ...
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Does anyone know if it's possible to solve $x^x=x+1$ in terms of $x$?
So I have tried solving for $x$ algebraicly using the productlog function but all I was able to do is:
$$x\log(x) = W(x\log(x)(x+1))$$
Maybe I could use the square-super root formula $e^{W(\log(x))}$, ...
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Is field of complex numbers closed under tetration?
The set of real numbers is closed under multiplication, but not under exponentiation (Eg. square root of negative numbers). That is, $\exists a, b \in R \mid {a^b} \notin R$. Then we introduced ...
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Can we generalize the tetration to the real or complex numbers? [duplicate]
is it possible to find a value for this operation: ${^{(3/2)}2}$?
If so, can we generalize the domain of the function ${^{x}a}$ to the real or even complex numbers?
I had originally tried to solve the ...
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Is there a proof showing super roots and super logarithms won't lead to a solution for the quintic?
So I am learning about tetrations and I just learned that tetrations are not elementary functions. When I heard that I remembered back to the statement that there is no general solution to the quintic ...
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Examples of closed forms of integrals with a power tower argument using W-Lambert function.
Here is a closed form of an integral that looks like:
Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$
...
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