If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...

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How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... ...
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Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation ...
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45 views

How to prove that a tetration holds the same last digit after a certain rank?

Let's say $u_{n} = 7^{u_{n-1}}$ and $u_{0} = 7$, how can we prove that for $n > 0$, that we will get the same last digit? I know that the last digit of $7^7$ is 3, we can prove it using ...
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1answer
59 views

Which one is greater?

For any $x\in\mathbb{R}^+$, let $x\diamond 1=x$ and $x\diamond (n+1) = x^{x\diamond n}$ for $n\in\mathbb{N}$. For example, $2\diamond 3 = 2^{2^2}=16$. If $t$ be an unique positive real number such ...
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How can I solve this equation $x^{x^{x^{x^{.^{.^{.}}}}}}-a=0$

I always use the Newton-Raphson Method if I want to find the roots of any equation as follow $$x_{1}=x_{0}-\frac{y_{0}}{y'_{0}}$$ But I don't know how to use this method if the equation takes the ...
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197 views

Finding solutions to $ x^x = 2x$

A friend claims it isn't possible to find a closed form for the smaller positive real solution of $x^x = 2x$. Numerically we have seen that $0.346...$ and $2$ are solutions, but are failing to do ...
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77 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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98 views

A peculiar observation about infinity.

Let ${\sqrt2^\sqrt2}^{\sqrt2^...}=y$. Then $\sqrt 2^y=y$ $\implies \sqrt 2=y^{1/y}$ $\implies \sqrt 2 =1$ $\implies 2 =1$ !! but how come that be. Can anyone explain this and point out what is ...
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How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
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40 views

Which primes satisfy this modular property?

Let $x$ be a residue$\mod p$ where $p$ is an odd prime. Im searching for such $p$ such that there exists a function $f(x)$ with propery $f(f(x)) - 2^x \equiv 0 \mod p $ for all values of $x$. I ...
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75 views

Tetrations of non-integers? [duplicate]

A Tetration is defined as $${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$$ or, by a recursion function, $${^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} ...
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78 views

Is there an inverse operation of tetration for values between 0 and .3?

So I don't understand everything with tetration, but the graph of $^2x$ or $x^x$ does not have any values between 0 and .3 on the y axis. So if we were to trying inverse tetration on say .2 what is ...
5
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1answer
141 views

How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
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422 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
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1k views

Explain $x^{x^{x^{{\cdots}}}} = \,\,3$

$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$ This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...
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2answers
145 views

Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
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1answer
105 views

Iteration of $x \to x^x$

If $u(x)=x^x$ then we can form $$ u^2(x) = \left(x^x \right)^{x^x} = x^{x^{x+1}} $$ some simplification occurs, but the further iterates are a typographical challenge to mathjax. Writing $E_k$ for ...
2
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1answer
60 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
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385 views

How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]

Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.
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1answer
89 views

Is it true that ${^\infty}{\sqrt[x]{x}} = x$

I was fiddling around with tetration and I stumbled across an interesting idea, ${^\infty}{\sqrt[x]{x}}$. I messed around with the concept a little bit and I had the following idea: Let ${^\infty}y = ...
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1answer
103 views

After Exponentials

Look at this- $$f(a,b)=a+b$$ The next step would be to make a function $g$ such that $$g(a,b)=\underbrace{a + a + a \cdots}_{b\text{ times}}=a\cdot b$$ Then we made $h$ so that ...
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74 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
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212 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
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237 views

Tetration graph for this function

I'm trying to visualize the derivatives of exponential tetration, by taking the original equation from its graph. Right now there is no elementary way of expressing the derivative. I'm not allowed to ...
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1answer
74 views

A question on the equation $^qx=2$

Given the equation $$^qx=2$$ with $q\gt3$ where $^qx$ means the 'tetration' operation on $x$, my question is: is it possible to find a value for $q$ for which the solution $x$ of the equation is a ...
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151 views

Efficient algorithm for calculating the tetration of two numbers mod n?

I'm trying to study the algebraic properties of the magma created by defining the binary operation $x*y$ to be: $ x*y = (x \uparrow y) \bmod n $ where $ \uparrow $ is the symbol for tetration. ...
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1answer
108 views

Is the exact value of $^{\infty}i$ known? [duplicate]

Wikipedia's article on tetration has a table of successive tetrations of $i$ that seems to imply that $^{\infty}i$ converges, and my own experimentation seems to confirm it, but I'm suspicious of ...
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344 views

If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$?

In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. ...
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179 views

Does infinite tetration of negative numbers converge for any value other than -1?

Okay, so I know that for positive values, $^{\infty}x$ converges to $-\frac{W(-\ln x)}{\ln x}$ for $e^{-e}\le x \le e^{\frac1e}$. Above that, it diverges. For positive values less than $e^{-e}$, any ...
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1answer
343 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
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117 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
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175 views

Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, … don't change, and generalisation

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
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120 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
4
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123 views

Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...
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435 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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1answer
162 views

Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
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1answer
157 views

Comparison between two tetrations

For a given natural number n, what ist the least number m, such that $$e \uparrow \uparrow m > \pi \uparrow \uparrow n$$ It seems that m = n + 1 is the desired number. Is this true for all n ?
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127 views

Is there a proof that the number $2 \uparrow \uparrow m+3 \uparrow \uparrow n$ is always squarefree?

I searched prime factors of the numbers $$z(m,n) := 2 \uparrow \uparrow m + 3 \uparrow \uparrow n$$ where $m,n\ge1$ Interestingly, z(3,3) is prime, the largest prime I found so far and probably the ...
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163 views

Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
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1answer
74 views

Does $a \uparrow \uparrow (n+1)-a \uparrow \uparrow n$ divide $a \uparrow \uparrow(n+2) - a \uparrow \uparrow(n+1 )$?

Does $$a\uparrow \uparrow (n+1) - a\uparrow \uparrow n$$ divide $$a\uparrow \uparrow (n+2) - a\uparrow \uparrow (n+1)$$ for all $a,n \ge2$ ? The case n = 0 is easy : $$a^a-a=a(a^{a-1}-1)$$ and ...
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135 views

Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than ...
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1answer
172 views

Stationary prime factor 34276387 , confirmation and generalization wanted.

Sheldon L found out, that $$34276387$$ is a prime factor of $$2 \uparrow \uparrow n + 3 \uparrow \uparrow n$$ for any natural number $n \ge 5$. $2 \uparrow \uparrow n$ leaves the remainder ...
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1answer
168 views

How do we know that tetration is exclusively right-associative?

When we go from multiplication to exponentiation we lose commutativity ($3^2 \neq 2^3$). Perhaps when we go from exponentiation to tetration every operation yields two possible results.
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317 views

Calculating $a^n\pmod m$ in the general case

It is well known, that $$a^{\phi(m)}\equiv1\pmod m ,$$ if $\gcd(a,m)=1.$ So, $a^n\pmod m$ can be calculated by reducing n modulo $\phi(m)$. But, for the tetration modulo $m$ $$a \uparrow ...
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276 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
5
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2answers
246 views

radius of convergence of half iterate of sinh(z)?

The half iterate of sinh(z) has a formal power series, centered around z=0. Does the formal power series for the half iterate converge at the origin? This is equivalent to asking if the half iterate ...
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1answer
127 views

Slight confusion about the calculation of the last digits of graham's number

In Wikipedia, Graham's number, it is described how to calculate the last d digits of Graham's number. They introduce an algorithm simply iterating $$x = 3^x \mod 10^d$$ d times starting with x=3. ...
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202 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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50 views

Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
23
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2answers
794 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...