Articles of tetration

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 ?

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 be partitioned into: $\{2,2\}$ $\{1,2,1\}$ But $\{2,1,1\}$ and $\{1,1,2\}$ are equivalent to $\{2,2\}$ No result containing $\{1,1,1\}$ should be enumerated since this […]

Is this a valid proof for ${x^{x^{x^{x^{x^{\dots}}}}}} = y$?

So I got this challenge from my teacher. Solve ${x^{x^{x^{x^{x^{\dots}}}}}} = y$ (eq. 1) for $x$. My attempt: As $x^{y^z}$ per definition equals $x^{y \cdot z}$, then $x^y = y$ from (eq. 1). Thus, $x = \sqrt{y}$. Is this a valid proof?

Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?

In the answering of another question in MSE I’ve dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic fixpoints. Another complex base on the unit circle however, $b = \sqrt{0.5}(1+i)$ gives only one fixpoint. I asked myself, whether there […]

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 $$z(14,15) = 14 \uparrow \uparrow 3 \ +\ 15 \uparrow \uparrow 3 = 14^{14^{14}}+15^{15^{15}}$$ turned out to be a candidate surviving trial division up […]

Algorithm for tetration to work with floating point numbers

So far, I’ve figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable b must be an integer number. How can I modify the algorithm so that both a and b can be floating point numbers and the correct answer will be produced? // […]

Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, some being labels such as: $2\times = \text{double}\\ \text{^}2 = \text{squared}$ And some that are also other functions, such as: $\text{^}0.5 […]

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 such a notation I’d like to write: $$ x ^ y = \times ^ y (x) $$ […]

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be any function that satisfies $$\lim\limits_{n\to\infty} \left(e^{\frac1e} + \frac1n\right)^{\wedge\wedge}\left[(10 n)^{1/2} + n^{A(n)} + C+o(1)\right] – n = 0$$ where $C $ is a constant. Then $\lim\limits_{n\to\infty} A(n) = \frac1e $ Conjectured by […]

How to calculate generalized Puiseux series?

I recently posted this, post containing a series of questions concerning the integration of ${x^{x^{x^x}}}$. In order to do so, I wrote ${x^{x^{x^x}}}$ as the following infinite summation: $$1 + \sum_{n=0}^{\infty}\sum_{k=0}^{n}\bigg(\frac{x^{n+1} \log^{n+1+k}(x)C_{kn}}{\Gamma(n+2)}\bigg)$$ I got to this summation using a “generalized Puiseux series”. I mention in the post that I am unsure of how to calculate […]