Articles of hyperoperation

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} (x_{n-1} \text{^} x_n)))$$ How about an $n$-ary tetration operator? $$\underset{i=1}{\overset{n}{\boxed{\underset{\leftarrow}{\LARGE{\text{4}}}}}}\, x_i = x_1 \uparrow\uparrow (x_2 \uparrow\uparrow (\cdots \uparrow\uparrow (x_{n-1} \uparrow\uparrow x_n)))$$ $$\underset{i=1}{\overset{n}{\boxed{\underset{\rightarrow}{\LARGE{\text{4}}}}}}\, x_i = (((x_1 \uparrow\uparrow x_2) \uparrow\uparrow \cdots) \uparrow\uparrow x_{n-1}) \uparrow\uparrow […]

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related to these functions, but I am not sure if they still keep the property of being NOT primitive recursive. My intuition is […]

How to extend this extension of tetration?

if $0\le b<1$, then $a↑↑b = a^b$ if $b\ge1$, then $a↑↑b = a^{a↑↑(b-1)}$ if $b<0$, then $a↑↑b = \log_a(a↑↑(b+1))$ so for example, $2↑↑\pi = 2^{2^{2^{2^{0.1415926…}}}} = 21.5963561$ How can I extend this to complex numbers? https://www.desmos.com/calculator/p8qcvngczb3

Algorithm for comparing the size of extremely large numbers

Is there a simple algorithm to decide which of the numbers $$a \uparrow ^b c \text{ and } d \uparrow ^e f$$ is the bigger one ? Using the hyperoperation, the numbers can be denoted with $$H_{b+2}(a,c)\text{ and } H_{e+2}(d,f)$$ I tried using the recursive definition of $H$ $$H_n(a,b) = H_{n-1}(a,H_n(a,b-1))$$ and induction to get […]

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 hyper-operations, as they are called. For example: Adding the forces on an object to find the resultant force. Multiplying the length, width, […]

Has this phenomenon been discovered and named?

If $$x-\frac{x}{2}=\frac{x}{2},$$ and $$\frac{x}{\sqrt{x}}=\sqrt{x},$$ and $$x-\uparrow(x-\uparrow^22)=x-\uparrow^22$$ when $(x\uparrow^n-[A])\uparrow^nA=x$, where $A$ is some constant, and one uses standard Knuth up-arrow notation[where ($\uparrow^{-1}x$)($-\uparrow^{-1}x$), ($\uparrow^0x$)($-\uparrow^{0}x$), ($\uparrow{x}$)($-\uparrow{x}$), ($\uparrow^2x$), and ($-\uparrow^2x$) all mean addition(and subtraction), multiplication(and division), exponentiation(and roots), and tetration(and super-roots) in that order],will $$x-\uparrow^n(x-\uparrow^{n+1}2)$$ always yield $$x-\uparrow^{n+1}2$$ for every degree $n$? If so, could someone please let me […]

Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next one $\epsilon_{n+1}$, the operation that does that is a left associative operation i.e. $${\tiny⋰}^{\epsilon_n^{\epsilon_n}}={}^{\omega}(\epsilon_n)=\epsilon_{n+1}$$ Since the exact same rule holds […]

Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I’m programming a maths environment. I’m computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were “invented” out of necessity to solve equations like $$ x^2 = -1$$ As far as I can tell, the imaginary units present in the split-complex […]

Could someone tell me how large this number is?

Context: If you guys didn’t know, I’m running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to join). If you are entering, do note that I am accepting entries for quite a while (ignore all […]

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, primarily by exploring various manipulations using logarithms and polylogarithms but have gotten nowhere. Although it is simple enough to show that $y(\sqrt{2})>y(1)$ and if $y'(x)>0$ for some […]