For questions relating to the computation, estimation and properties of extremely large quantities that are not usually used in mainstream mathematics.

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3answers
90 views

What is the Googol root of a Googolplex? [on hold]

$\text{Googol}=10^{100}$ $\text{Googolplex}=10^{\text{Googol}}=10^{{10}^{100}}$ What is $\sqrt[\text{Googol}]{\text{Googolplex}}$? I know that's the same as $\sqrt[10^{100}]{10^{10^{100}}}$ but I ...
1
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2answers
83 views

Is this the correct way to compute the last $n$ digits of Graham's number?

For the following question, all what is needed to know about Graham's number is that it is a power tower with many many many $3's$ Consider the following pseudocode : input n Start with $s=1$ and $...
3
votes
2answers
207 views

What are the last 20 digits of mega?

What are the last 20 digts of the number mega, which is "2 in a pentagon" in steinhaus-moser-notation ? In contrary to power towers or tetration, the ending digits are not stable. I found out that ...
2
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2answers
151 views

How does one prove that $(2\uparrow\uparrow16)+1$ is composite?

Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: $$F_1=2+1\...
0
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1answer
75 views

Large, small but a useful number. [closed]

Today we were discussing in our class about usefulness of a number no problem how large,small may be it's value. As per my knowledge (till grade 11) Avogadro number $N_A=6.022\times 10^{23}$ is a ...
2
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1answer
55 views

How can I show, that $N\uparrow\uparrow N$ is not “much larger” than $N$ for very large $N\ $?

Here : https://sites.google.com/site/largenumbers/home/3-2/knuth Saibian demonstrates that for very large numbers $N$, $N\uparrow\uparrow N$ is only "slightly larger" than $N$. I would like to ...
3
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1answer
67 views

explicit upper bound of TREE(3)

TREE(3) is the famously absurdly large number that is the length of a longest list of rooted, 3-colored trees whose $i$th element has at most $i$ vertices, and for which no tree's vertices can be ...
1
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1answer
73 views

How can I prove this sharp upper bound?

Here : What are sharp lower and upper bounds of the fast growing hierarachy? Deedlit mentions that for natural $m,n\ge 2$ and natural $k>n+log_2(n)$ , we have $$2\uparrow^{m-1}n<f_m(n)<2\...
2
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2answers
137 views

Where does googolplum lie in the fast growing hierarchy?

Here : https://sites.google.com/site/largenumbers/home/3-2/andre_joyce Saibian presents the largest number coined by Andre Joyce, googolplum. It should lie at the $f_{\omega+2}$-level in the fast ...
6
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4answers
145 views

How do we compare the size of numbers that are around the size of Graham's number or larger?

When numbers get as large as Graham's number, or somewhere around the point where we can't write them as numerical values, how do we compare them? For example: $$G>S^{S^{S^{\dots}}}$$ Where $G$ ...
0
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1answer
108 views

What is the smallest number $n$ , such that $n\uparrow^4 n>3\uparrow^5 3$ holds?

What is the smallest number $n$, such that $$n\uparrow^4 n>3\uparrow^5 3$$ holds ? $\uparrow$ stands for Knut's up-arrow-notation and is defined as follows $a\uparrow b=a^b$ $$a\uparrow \...
4
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1answer
156 views

Comparing up-arrow's

Is it true that $$3\uparrow^{n+1} 3\ >\ n\uparrow^n n $$ holds for every $n\ge 1$ Since $3\uparrow^{n+1}3=3\uparrow ^n 3\uparrow ^n 3$ and $3\uparrow^n3$ is much bigger than $n$ for $n\ge 3$, ...
3
votes
3answers
3k views

Find modulo of multiplication of two number?

Given $m$, $a$ and $b$ are very big numbers, how do you calculate $ (a*b)\pmod m$ ? As they are very big number I can not calculate $(a*b)$ directly. So I need another method.
3
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3answers
108 views

The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
3
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1answer
187 views

Proof of the inequality $2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$

I tried to prove the inequality $$2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$$ for all natural numbers $n\ge 1$ For $n = 1$, the claim is true because of $16 < 27 < 32.$ The left ...
2
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1answer
55 views

What is the smallest number $n$ for which $bb(n)>f_{\epsilon_0}(5)$ is known?

It is known that $bb(23)$>Graham's number (I do not remember exactly, but $bb(21)$ could already be larger). But what is the smallest number $n$, such that $bb(n)>f_{\epsilon_0}(5)$ is known ? ...
24
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8answers
842 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
3
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0answers
84 views

Does Graham's number have an odd or an even number of digits?

I think it is hopeless to decide whether the number of digits of Graham's number is even or odd because the only way that I can think of is determining the logarithm with accuracy $0.1$ or even better,...
0
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1answer
44 views

Extending fast growing functions to the reals “naturally”

There are a lot of incredibly fast growing functions defined on the natural numbers. Typical examples start with tetration, further hyper operators, Ackermann, and then there is monsters like the ...
8
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2answers
2k views

What's the last whole number before a googolplex?

"What's the last whole number before a googolplex?" My six yr old asked me this tonight. How does a math challenged dad answer this?! A googolplex is hard enough as it is to imagine or visualize. ...
-1
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2answers
139 views

Exponentiation and far too high numbers?

I love very, very, very, big numbers! You see, I'm working on powers of $2$ and I need to calculate the next expression in this sequence: $2^2=4$ $2\uparrow\uparrow2=216$ $2\uparrow\uparrow\uparrow2=...
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7answers
2k views

Is there a way to calculate absurdly high powers? [closed]

Could it be at all possible to calculate, say, $2^{250000}$, which would obviously have to be written in standard notation? It seems impossible without running a program on a supercomputer to work it ...
1
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2answers
56 views

In a given range how can i find how many times a two digit number appears ?

I want find how many times a two digit number appears in a given large range , Range is 10^500 . Example : I want to find 21 in given range and the range is 15 to 240 , there are total of 12 numbers ...
14
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1answer
1k views

Graham's Number : Why so big?

Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !
2
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1answer
118 views

How do I write Grahams number

I found that graham's number is :enter image description here So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?
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2answers
42 views
70
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5answers
6k views

Is it possible to represent every huge number in abbreviated form?

Consider the following expression. 16313107343153908912074032799466965289077771751767944648966669091376847859711382649033004075188224 This is a 98 decimal digit number. This can be represented as $...
6
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1answer
2k views

If I call the Ackermann Function with Graham's number as both of its arguments will it be less than $g_{65}$

In xkcd comic 207 it states that [xkcd] means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians. $A(g_{64},g_{64})$ In this explanation it ...
4
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0answers
78 views

What is the smallest prime factor of the number $14^{14^{14}}+13\ $?

What is the smallest prime factor of the number $$N\ :=\ 14^{14^{14}}+13\ ?$$ The number of digits of $N$ is $12,735,782,555,419,983$ (The number of digits of $N$ has itself $17$ digits). The first ...
0
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1answer
30 views

Big-O complexity of $2t(\frac{n}{2}) + n^3$

I'm trying to determine the Big-O complexity of the listed equation and want to know if my approach is valid. I tried using the Master method. It appears to be a case $3$ type problem to me, where $f(...
0
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1answer
69 views

How many values within a range have a base-256 representation that contains all the digits in a set?

A hobbyist programmer enquires... ** Situation: ** An iterator iterates over a range of big-numbers from 'min' to 'max'. The current iteration's value is represented by a fixed-length array of ...
1
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2answers
97 views

efficient way to express large numbers

I recently watched the walkthrough of Graham's Number on YouTube (Numberphile). Mind-blowing of course. I then puttered around in other large number topics like Ackerman and Tree(3) and fast growing ...
3
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3answers
247 views

Graham's Number versus another large number

I recently read this article about the most damage you can do in a single turn in Magic the Gathering. According to the current version of the deck, that damage is about a) $2 \rightarrow 17 \...
2
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0answers
57 views

Can I restrict the possible factors of $2\uparrow \uparrow 4+3\uparrow \uparrow 4$?

I would like to accelerate the search of prime factors of $$2\uparrow \uparrow 4+3\uparrow \uparrow 4$$ In a question, I asked for a prime factor and another user also asked, whether this number is ...
1
vote
2answers
53 views

How to shuffle a number so that it can be maximum multiple of the number 30 ?

If i have a large number (<=10^5 Digits) how can i tell that if i can shuffle the number so that it become a multiple of 30 . if it is possible then i have to find the maximum multiple . Suppose if ...
3
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2answers
833 views

Which number is larger?

If $n$ is large enough, which is greater: $(n+1) ^{n+1}$ or $(kn)^{n}$ where $k$ is a natural number? I've plotted a graph which suggests that the second is larger, but surely the larger power ...
33
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4answers
2k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
4
votes
2answers
204 views

With $f(n) = n!$, what is the least $k$ such that $f^k(\text{googolplex}) > \text{Graham's number}$?

$\text{googolplex} = 10^{(10^{100})}$ Is $\text{googolplex}!$ greater than $\text{Graham's number}$? How would this be proven? If $\text{googolplex}! \le \text{Graham's number}$, (which I expect) ...
3
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1answer
84 views

Can different tetrations have the same value?

Suppose, we have two numbers $a\uparrow \uparrow b$ and $c\uparrow \uparrow d$. To avoid trivial cases, suppose $a,b,c,d>2$ and $(a,b)\ne (c,d)$. Is there a quartupel $(a,b,c,d)$ with $a\...
0
votes
1answer
88 views

Bowers array notation : $f_{\omega^\omega}(n)\approx [n,…,n]$ ($n$ times)

I learnt at this site that $$\large f_{\omega^\omega}(n)\approx \underbrace{[n,...,n]}_{n\ n's}$$ For a simular approximation $$\large f_{\omega^2}(n)\approx \underbrace{n\rightarrow n\rightarrow......
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1answer
118 views

Jonathan Bowers' multidimensional arrays

Sbiis Saibian designed a site with large numbers (first hit with the search Sbiis Saibian) In section $4.1$ he describes Bowers' notation, but unfortunately he did not come to the multidimensional ...
0
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0answers
58 views

For which natural numbers $m,n>1$ does the inequality $2\uparrow^m n>f_m(n)$ hold?

Denote $$f(n,m):=2\uparrow^{m-1} n$$ (See : Wiki ) and $$g(n,m):=f_m(n)$$ (See : Wiki ) It is straightforward to show $f(n,m)<g(n,m)$ for all $m,n>1$ via induction. But for which $m,n>1$ ...
2
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0answers
93 views

Compute sum of large powers [closed]

I have the following problem. There is an array that contains values that are to be powers of $-2$. I need to calculate the sum of these powers. For example, if the array is $\{3,4,5\}$ I need to ...
2
votes
2answers
31 views

The number of logarithm applications to get from n below 1

Let $L(n)$ to be a number of logarithms that you need to apply on $n$ until you get below 1: $$ 0 \leq \log\cdots\log n < 1 \\ \uparrow \\ L(n)\mbox{-times} $$ Is there a name for this function? ...
0
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1answer
81 views

Tight bounds for Bowers array notation

This link http://googology.wikia.com/wiki/Array_notation shows the definition of bowers linear array notation and the approximation $$\{n,a+1,b+1,c+1,d+1,...\}\ \approx f^a_{...+\omega^2d+\omega c+b}...
3
votes
2answers
70 views

Quick Exponent Clarification

$N = 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5}}}}$ In the following equation is N equal to $5^{5^4}$ or $5^{(5^{(5^{(5^5)})})}$? One of them is huge compared to the ...
0
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2answers
244 views

How can the number $\left\langle \matrix {3&3\\3&3}\right\rangle $ be described?

http://qntm.org/planar shows how Jonathan Bowers defines numbers using $2D$-arrays. I would like to get a feeling how big such numbers are. How can the number $$\left\langle \matrix {3&3\\3&...
2
votes
2answers
97 views

Do “small” and “large” numbers actually exist in an absolute sense?

Numbers like $(10)^{-10^{10^{10}}}$ are generally regarded as small, whereas numbers like, for example, Graham's Number, are regarded as extremely large. My question is, are these numbers simply "...
8
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5answers
582 views

Is there a number so large that we could never calculate it?

Note that I edited this post significantly to make it more clear (as clear as I think I could possibly make it). First, let me mention what I am NOT asking: I am NOT asking for the largest number we ...
1
vote
1answer
55 views

Comparing conway chains

See https://en.wikipedia.org/wiki/Conway_chained_arrow_notation for the details how conway chained arrow notation works. I want to calculate the approximate value $n$ such that $$n\rightarrow n\...