Questions tagged [hyperoperation]

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

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Notation for Expressing Hyperoperations and their Inverses

This is a table for the new mathematics notation alongside its equivalent old mathematics rendition: \begin{align} a *_1 b & \equiv a + b & & a \; /_1 \; b \equiv a - b & & a \; \...
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Is $0$ the Exponential Inverse?

For a while, I've been wondering why this pattern seems to allude to the fact that $0$ is one of the inverses of exponentiation. \begin{align} & x \cdot -1 = -x & \text{Inverse of addition} \\ ...
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hyperidentities, unknown evaluation

I do not follow the notion of hyper-identities very clearly. In the last line in the snippet, how looks the substitution $$x_1x_2x_1$$ for $F$, what is the result and can I see that this is not ...
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Why does 2 result in the same value regardless of whether it is added to itself, multiplied by itself, or put to the power of itself? [closed]

I'm inferring that any hyperoperation you could apply here using two for every value would result in four. Why is this?
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Why are tetrations not useful?

I've always wondered after learning addition, multiplication, and power facts (and their inverse operations) what the next higher level of facts I would need to memorize would be. However, instead of ...
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Is anything significant known about inverting power towers?

This has to be something somebody has looked into, but I can't find it. In general, for finite expressions of the form $${a_1}^{{{a_2}^{{⋰}^{a_i}}}},$$ if one reverses the usual order of ...
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Closure under reverse tetration

The natural numbers are closed under addition, but not subtraction The integers are closed under multiplication, but not division The rationals are closed under exponentiation, but not roots The real ...
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What does $i\uparrow\uparrow i$ equal to?(or, it has no result?) [closed]

What does $i\uparrow\uparrow i$ (or, i↑↑i) equal to? $i+i=2i$ $ii=-1$ $i\uparrow i=i^i≈0.20787957635076193$
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What is the series of identities for these hyper-operations?

Suppose we define the hyper-operations iteratively off the previous hyper-operation. We begin with the zeroth hyperoperation being addition, $$a\circ_0b:=a+b.$$ We then assert that $$a\circ_{n+1} b:=\...
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Does anything precede zeration in the hyperoperators

"Hyperoperator" family of operators is defined recursively like so: $$ a\langle{0}\rangle{b} = a+b\\ a\langle{1}\rangle{b} = ab = a+a+a+a... \; \text{(w/ b a's)}\\ a\langle{2}\rangle{b} = a^{...
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Constant difference symbol

A simple notation question: I'm familiar with the notation $a \propto b$, which means $a$ and $b$ are constant multiples of each other, but is there an analogous symbol for when $a$ and $b$ differ by ...
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How are the hyperoperations defined?

For the first few operations such as addition and multiplication they follow rules such as $$A(x,y+1)=A(x,y)+1$$ $$A(x,0)=x$$ For multiplication $$M(x,y+1)=M(x,y)+x$$ $$M(x,1)=x$$ So for a general ...
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Solutions to equations involving hyperoperations

Are there any texts on solutions to equations involving hyperoperations? Let's define $H_n(x,y) : \mathbb{N}_0 \rightarrow \mathbb{N}_0 $ to be the n-th hyperoperation, in particular: $H_n(x,y)=\...
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Compare the growth rate of $f(x)$ and $g(x)$ and show how much quicker the quicker of the two grow

Which of the 2 grows quicker? $f(x)$, $g(x)$, or both grow equally as quick? Definitions Notation $a [x] b = a ↑^{x-2} b$ Functions $f(x) := \begin{cases} 1, & \text{if } ...
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Show the original Ackermann function is non-primitive recursive

There are a couple of questions here which show that the modern Ackermann function $A(i, x)$ is not primitive recursive. This new Ackermann function defined by Péter is a simplification of the ...
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Inverse hyperoperation code library

I hope you're having a nice day, I am an Electrical Engineer and I am trying to do a math paper, but I need a coding library in any coding language that can do inverse hyperoperations, do you know of ...
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"Shooting Room" ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
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We know that $2+2 = 2 \times 2 = 2^2$. Is this true for all successive hyperoperations?

Given that multiplication is repeated addition and exponentiation is repeated multiplication, if we were to continue onwards, what would the result be for successive hyperoperations? If we take the ...
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"Multi-distributive" operations on sets.

As you can remember from Algebra courses, if $A$ is a set, and exist functions $\oplus , \odot: A \to A$ such that $(A,\oplus,\odot)$ is a ring, then such ring has a distributive property such that $$\...
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Using pentration with non-integer values: how to solve 2^^^1.5? [duplicate]

Say we define using hyperoperators with non-integers the same way we do with exponentiation, that is to say we convert the 'exponent' to a fraction and raise the base to the hyper-power of the ...
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Tetration: Summation of $\displaystyle {1 \over{}^{n}2}$

I'm going to get straight to point with this question - Can you find a closed form solution to this sum. $$\sum_{n=1}^\infty \displaystyle {1 \over{}^{n}2}$$ (where ${}^{n}2$ represents the nth ...
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Are there more identity numbers past 2?

I apologize in advance if this question is too ill-posed, but here we go. As an additive identity, $x+0=x$. As a multiplicative identity, $x\times 1=x$. $2$ feels similar in a way I can't define as ...
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New commutative hyperoperator?

After reading about Ackermann functions , tetration and similar, I considered the commutative following hyperoperator ? $$ F(0,a,b) = a + b $$ $$ F(n,c,0) = F(n,0,c) = c $$ $$ F(n,a,b) = F(n-1,F(n,a-...
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Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
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Fractional and complex rank hyperoperations [duplicate]

The first hyperoperation $H_1(a, b) = a + b$ is addition, and is defined over the complex numbers. The second hyperoperation $H_2(a, b) = ab$ is multiplication, and is also defined over the complex ...
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Non integer Hyper-powers. [duplicate]

If I have a function $y=x^x$, that can be denoted in hyperpower notation as $^2x$, but I will be denoting it as $ y= $ hyp$_2(x) $. In general, for hyperpowers, $y=x^{x^{x^{...}x}}$ or in my notation $...
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Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is ...
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Tetrating by non-integers? [duplicate]

Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (x[4]x), which is the same as x pentated by 2 (x[5]2), would look like on a graph. To do this, ...
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A pre-successor hyperoperation: subplus

I wrote up these notes about a month ago and just found them. I could really use a second opinion or two. I notice there's a whole question about how my whole thesis is wrong, but that doesn't ...
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Does $2^{k!}$ have any significance w.r.t. primality or Wilson's Theorem?

Typically, Wilson's is given as $$(n-1)! \equiv -1 \pmod{n},$$ which is short and sweet, but I came up with an alternate presentation of it that's arguably needlessly complicated, but also arguably ...
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Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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Why order matters in hyperoperations after exponentiation? [duplicate]

In first and second hyperoperations (addition and multiplication), order of two operands doesn't matter, result is same. However, it doesn't work on exponentiation. Why it does matter in ...
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Domain for Knuth's up-arrow

The wikipage for Knuth's up-arrow notation says $a \uparrow ^n b$ is defined recursively for integer $a$ and non-negative integers $b$ and $n$ like so: \begin{equation} a\uparrow^n b = \begin{cases}...
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What Tetration Algorithm can I utilize

I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow": A mandelbrot fractal (I call it "operation order 1") is done by ...
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How to compute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $ \int{x↑↑n} dx $ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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A new notation for operations and some questions [duplicate]

let $x^{/1/}=x+x$ (addition) $x^{/2/}=x.x$ (multiplication) $x^{/3/}=x^x$ (exponentiation) $x^{/4/}={}^xx$ (tetration) and so on..... I have the following questions: $(1)$ Can we define an ...
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Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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Extending Knuth up-arrow/hyperoperations to non-positive values [duplicate]

So... I had the silly idea to extend Knuth's up-arrow notation so that it included zero and negative arrows. It is normally defined as $$\begin{align*} a \uparrow b & = a^b \\ a \uparrow^n b & ...
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Why do we have the present order of operations, and how do hyperoperations fit in?

Something that's been bugging me for a fairly decent while is the order of operations - not so much using it, however, as to understanding where it comes from. Typically we're introduced to it in the ...
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What is the name for this: "x^^y" (as in 2^^2=2^2 and 3^^3=3^3^3 and so on)

How do you call / is there any specific name for the following: x^^y e.g.: 2^^2 = 2^2 3^^3 = 3^3^3 4^^4 = 4^4^4^4 ...
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Under what "natural" combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher ...
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Tetration by a Non-Integer

Does anyone think that tetration by a non-integer will ever be defined ... really properly? Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time ...
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Can we evaluate noninteger hyperoparations?

The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on. What happens when $n$ is noninteger? Can we evaluate, e.g. $...
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Could there be an $\omega_1^{CK}$th hyperoperation? [closed]

If addition is the first hyperoperation, multiplication is the second, and the $(\alpha+1)$th hyperoperation is repeated occurrences of the $\alpha$th one. Is it possible for a limit ordinal (for ...
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Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
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Can there be fractional hyper operators and if so how to they function? [duplicate]

If we call addition (and subtraction as they are the same) the first hyperoperator, multiplication (and division) the second, exponentiation (radicals, indices and logarithms) the third, then ...
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How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?

3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms ...
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Question about $TREE(3)$ and Graham's Number

Lets say: $G=\text{Graham's Number}$. And: $$ \begin{align*} \alpha_1&=G\uparrow^G G, \\ \alpha_2&=\alpha_1\uparrow^{\alpha_1} \alpha_1 \\ &\vdots\\ \beta_1 &= \alpha_G\uparrow^{\...
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Hyperoperation: Why does $H_n(0,b) = 0$ for $n\ge4$, $b$ odd ($\ge -1$)

On the Wikipedia for hyperoperations is says that $H_n(0,b) = 0$ if $n\ge4$, $b$ odd ($\ge -1$). From what I gether, the basic jist of what this is saying is that $0^{0^0}=0$ and from that we can ...
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Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed? [duplicate]

Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper): $$ \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...