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 $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [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
52 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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2answers
59 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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0answers
84 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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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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26 views

Are these correct for calculating the number of Digits

I think the following equations are correct for base 10, where each D corresponds to the number of digits in results for each operation. Dadd = Floor(Log10X) + 1 Dmul = Floor(Log10X + Log10Y)+1 ...
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1answer
47 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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0answers
37 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
75 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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4answers
276 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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1answer
65 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
137 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 ...
4
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2answers
104 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 ...
2
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0answers
82 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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0answers
107 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 ...
4
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0answers
239 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 ...
9
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1answer
125 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
115 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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55 views

Two more numbers involving tetration with unknown prime factors

I have two more numbers for which I search prime factors : $$6 \uparrow \uparrow 4 + 7 \uparrow \uparrow 4$$ and $$7 \uparrow \uparrow 4 + 10 \uparrow \uparrow 4$$ I searched upto $10^9$ and did ...
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0answers
100 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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1answer
108 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
59 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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68 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
146 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 ...
0
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1answer
105 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.
4
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2answers
225 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 ...
6
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0answers
221 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
votes
2answers
144 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
81 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. ...
3
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0answers
176 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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0answers
38 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 : ...
14
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2answers
454 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 ...
4
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2answers
162 views

Which precision would be needed?

According to Wikipedia, it is not known whether the number $$\pi \uparrow \uparrow 4$$ is an integer. (See Tetration) To which precision would $\pi$ have to be calculated to decide this ? The ...
20
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1answer
358 views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
3
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2answers
192 views

Knuth's arrow up notation again

Consider the following recursion : $$ a(1)=3! $$ $$ a(n+1) = a(n)! \quad\hbox{for all $n\geq 1$} $$ For a given $n$, how can the number $m$ with $$ 10 \uparrow \uparrow m < a(n) < 10 \uparrow ...
4
votes
1answer
185 views

Knuth's up arrow notation

Given n, how can the number m with $10 \uparrow \uparrow m < 2 \uparrow \uparrow n < 10 \uparrow \uparrow (m+1)$ be calculated ? With induction, I got $10 \uparrow \uparrow m < 2 ...
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0answers
34 views

Is crossdirectional partial tetration of order $n^c$?

The following animation shows a sum of a matrix where the parameter $c=0$ gives a straight line in pink, and as $c \rightarrow 1$ it approaches the Chebyshev $\psi$ function, the blue staircase. This ...
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2answers
60 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
0
votes
1answer
90 views

Does this follow from the definition of the LambertW function?

The LambertW function $W(s)$ also called ProductLog seems to satisfy this relation: $$-W(s) = \underbrace{-s e^{-s e^{\cdot^{\cdot^{-s e^{-W(s)}}}}}}_n$$ Or truncated: $$-W(s) = -se^{-s e^{-s e^{-s ...
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0answers
69 views

Arrow notation and decimals.

We know how to add with decimals. We know how to multiply with decimals. We know how to exponentiate with decimals. Do we know how to work with decimals for power towers? for example, can we deal ...
2
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1answer
229 views
2
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1answer
184 views

Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
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0answers
26 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton ...
3
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0answers
108 views

Does $^{\frac12}x=e^{W(\ln x)}$, or not?

Whenever I see tetration discussed here, I inevitably see it asserted that there's no consistent continuous definition for tetration. However, it seems to me that If we restrict ourselves to ...
4
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2answers
278 views

Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
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2answers
234 views

Notation for function $ + \rightarrow \times $

Is there a standard notation to represent the function building the multiplication from the addition (I'm talking of the usual $+$ on $\Bbb N$)? I'm tempted by: $$ x \times y = + ^ y (x) $$ With ...
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0answers
110 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $f_1(x)=x,f_2(x)=x^x,\dots f_{n+1}(x)=x^{f_n(x)}~(n\geq 1).$ Let $F_n(x)=f_n^{'}(x).$ Hence $F_1(x)=1, F_2(x)=x^x(1+\log(x))\dots.$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
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1answer
143 views

Fixed Point of $x_{n+1}=i^{x_n}$

For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$. EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$ My ...
87
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1answer
20k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
5
votes
3answers
732 views

Calculating 7^7^7^7^7^7^7 mod 100

What is $$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$ I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.