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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47 views

What is the highest number that could be written down in principle using all computers in the world?

I read somewhere in the internet, that the capacity of all computers in the world would be about $10^{18}$ bytes. Does this mean, that in principle, a number with $10^{18}$ digits could be written ...
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0answers
14 views

Proof of Polyates Lemma

In Sbiis Saibian's site I came across Polyates Lemma which states that $$(b \uparrow^k m) \uparrow^k n\ <\ b\uparrow^k (m+n)$$ for all positive integers b,m,n,k with $b\ge 2$ and $k\ge 2$. He ...
3
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0answers
51 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 $n\ge 3$, ...
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0answers
38 views

Growth rate of $f(f(n))$, where $f(n)$ is the ackermann-function.

Let $$f(n)\ :=\ n \uparrow^n n$$ and $$g(n)\ :=f(f(n))\ =\ f(n)\uparrow ^{f(n)} f(n)=n\uparrow^n n \uparrow^ {n \uparrow ^n n} n\uparrow ^n n$$ So, $g(n)$ is $f(n)$ applied twice. What is the ...
5
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2answers
139 views

Approximation of (n^n)^n

To be specific, what is the best way to calculate the first 10 digits decimal approximation of $$ \large \left(123456789^{123456789}\right)^{123456789}$$? Even WolframAlpha gives the result in a ...
12
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1answer
546 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 !
6
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1answer
239 views

What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the ...
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0answers
20 views

Comparison of $4$-entry-conway-chains and $3$-entry-conway-chains

How big must $n$ approximately be, that $$n\rightarrow n \rightarrow n\ \approx \ 3 \rightarrow 3 \rightarrow 3 \rightarrow 3$$ holds ? $n\rightarrow n \rightarrow n$ (conway-chain) is ...
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0answers
28 views

Little graham's number, Graham's number and the Graham-conway-number

Sbiis Saibian desbribes on his site in section $3.2.9$ the "little-graham-number" He claims that Graham used this number (much smaller than "Graham's number") in his proof, and Gardner published ...
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0answers
23 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 ...
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1answer
323 views

Magnitude of $10 \uparrow \uparrow \uparrow 10$

Is there any way to understand the magnitude of $$10 \uparrow \uparrow \uparrow 10$$ ? I know that the number can be constructed as follows : $$M_1 := 10$$ $$M_2 := 10^{10^{10^{...10}}}$$ ...
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2answers
610 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 ...
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0answers
42 views

Approximation with conway chains

Suppose, we have the functions $$f(n)=n \uparrow^n n$$ $$g(n)=f^{f(n)} (n)$$ $$h(n)=g^{g(n)} (n)$$ My goal is to estimate the magnitude of $h(4)$ with conway chains (In the way, Graham's number is ...
23
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6answers
529 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 ...
6
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4answers
4k views

How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
2
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1answer
139 views

First digits of extremely large numbers (Generalization of “first digits of Graham's number”)

I found a question about the first digits of Graham's number and would like to generalize it : We want the first n digits of the number $a\uparrow^b c$. Which method is the most effective to do ...
3
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1answer
19 views

Predicting how long the result of a permutation

We're calculating the result of a 'tweaked' Birthday Problem, but when we're calculating, we stumped by a very nasty permutation. $$10^{576}P_{10^{16}}$$ Which, make us stop working at the number, ...
2
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2answers
171 views

Making sense of combinatorics-based marketing hyperboles

Diablo 3 has 97 billion possible skill/trait builds. Per class. LessPop_MoreFizz, emphasis is mine. I used base two logarithm to claim "97 billion" configurations only are roughly 37 binary ...
0
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1answer
82 views

Divisibility of huge numbers

Please help me to solve my homework ;) Prove that for any positive integer $n$ a square of rather big number divides even more huge number: $${\LARGE \left.\underbrace{33\dots 3}_{1\underbrace{00\dots ...
2
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1answer
284 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 ...
2
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2answers
270 views

How can I calculate or think about the large number 32768^1049088?

I decided to ask myself how many different images my laptop's screen could display. I came up with (number of colors)^(number of pixels) so assuming 32768 colors I'm trying to get my head around the ...
4
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0answers
95 views

Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
7
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4answers
7k views

What is the biggest number ever used in a mathematical proof?

Probably a proof (if any exist) that calls upon Knuth's up-arrow notation or Busy Beaver.
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1answer
37 views

Calculation of $f_{\omega^3}(2)$ in the fast growing hierarchy

How is the number $$\large f_{\omega^3}(2)$$ in the fast growing hierarchy calculated ? My only idea is to convert to $$\large f_{\omega^2 2}(2)$$ but now I have no idea how to continue.
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0answers
50 views

Monotony of the function $a\uparrow ^b c$ in all the arguments

I want to prove the following monotony properties of hyper-operations : 1) $ a \uparrow^b c < (a+1) \uparrow^b c$ for all $a,b,c \ge 1$ 2) $ a \uparrow^b c < a \uparrow^b (c+1)$ for all $a\ge ...
2
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2answers
91 views

For which n does the inequality $2 \uparrow^{n+1}n > 3\uparrow^n 3 +2$ hold?

For which n does the following inequality hold ? $$2 \uparrow^{n+1}n > 3\uparrow^n 3 + 2$$ where $\uparrow$ stands for knuth's up-arrow notation. I need this inequality to prove that ...
3
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1answer
116 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 ...
0
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1answer
42 views

Find the biggest number from given data below

Maths problem : $$(9^{62773} + 2)^{83721}$$ Now here is the rest of the problem. After finding the huge number I have to find its digital sum. If you don't know what that means just give me the ...
2
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1answer
151 views

Definition of the function $f_{\epsilon_0}$ in the fast-growing hierachy

I found an article where the growth of $$f_{\epsilon_0}$$ in the fast-growing-hierachy is described, but the text is very long and difficult to understand. Is ...
3
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3answers
229 views

Comparing $\large 3^{3^{3^3}}$, googol, googolplex

How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$). Thanks much in advance!!!
6
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1answer
853 views

Graham's number expressed using xkcd's “Knuth Paper-Stack Notation”

The title text for xkcd #1162 describes a method for expressing extremely large numbers: Knuth Paper-Stack Notation: Write down the number on pages. Stack them. If the stack is too tall to fit in ...
6
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2answers
209 views

Logarithms of logarithms of Graham's number, is the result ever handy?

The other day I was asked how to represent really big numbers. I half-jokingly replied to just take the logarithm repeatedly: $$\log \log \log N$$ makes almost any number $N$ handy. (Assume base ...
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0answers
86 views

Tim Chow's proof that the moser-number is much smaller than grahams number

The link here shows a proof from Tim Chow that the moser-number is much smaller than grahams number. I do not understand the inequality 3^^...^^3 (3^^^^^3×2-1 arrows) << G 2 What does G 2 ...
3
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2answers
1k views

How to explain Real Big Numbers?

Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the ...
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0answers
51 views

Bounds for $(10 \uparrow \uparrow 257) \uparrow \uparrow \uparrow (10 \uparrow \uparrow 257)$

A lower bound of 2[6] (Steinhaus-Moser-Notation) is $$ M:= (10 \uparrow \uparrow 257) \uparrow \uparrow \uparrow (10 \uparrow \uparrow 257)$$ I would like to bound M in the following way : $$10 ...
0
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0answers
28 views

Bounds for a sequence and the steinhaus-moser-numbers

Let $(a_k)$ be the following sequence $$a_1=n\ ,\ a_{n+1}=a_n^{a_n}\ for\ all\ n \ge 1$$ It is easy to show $a_k > n\uparrow \uparrow k$ for all $k > 2$. But is it true that $a_k < n ...
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0answers
41 views

monotony for conway chains

Let X and Y be chains, m and n be natural numbers with 0 < m < n. Is it always true that $$X \rightarrow m \rightarrow Y < X \rightarrow n \rightarrow Y$$ ? I need this to bound the ...
4
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1answer
3k views

Large numbers in real world

I am a high school math teacher and I am looking for a comprehensive list of large numbers which occur in real world. For example There are $10^{14}$ cells in the human body $10^{100}$ is called ...
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1answer
120 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 ...
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0answers
45 views

Potential values of minimum cost maximum flow algorithm

I have a simple directed graph $G(V,E)$ that has a source $s$ and sink $t$. Each edge $e$ of $G$ has positive integer capacity $c(e)$ and positive integer cost $a(e)$. I am trying to find the minimum ...
1
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1answer
112 views

Modular equation with very large powers

I studying for a discrete mathematics exam and am stuck on this question: Find the value of the unique integer $x$ satisfying $0 \le x < 17 $ for which: $$ 4^{1024000000002} ≡ x \pmod{17} $$ I ...
3
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0answers
368 views

First $n$ digits of Graham's Number

I know using Euler's Totient function, it's easy to find the last $n$ digits of Graham's number (or any large repeating power tower), but is there any known way to find the first $n$ digits of ...
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3answers
2k 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.
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3answers
204 views

What is the tens digit of $3^{100}$?

Is there a general formula to calculate the n-th digit of any big number?
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2answers
112 views

How to estimate the size of a ratio with very large factorials?

I want to estimate the size of the following ratio: $$\frac{10^{18}!}{10^{14}!\ 10^4!}$$ Since I don't have an idea how to simplify it and no CAS is able to handle numbers of this size, I am at an ...
2
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0answers
64 views

Improving Montgomery product

I am reading the paper "A Cryptographic Library for the Motorola DSP56000" (http://link.springer.com/content/pdf/10.1007%2F3-540-46877-3_21.pdf) which describes a trick to speed-up calculation of the ...
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4answers
75 views

Big Oh from function

I'm having a lot of trouble finding big oh for the function: $$f(i)=i+2i+\cdots+i\cdot i,$$ where $f(i)$ is the steps to run this function. Could you give me a hint as how to find it? Much ...
1
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1answer
199 views

Find the largest divisor of an integer $b$.

I want to find out an efficient method to calculate the largest divisor of a very big integer $b$ which can be up to $\large 2^{1000}$. That is, I want to find out an integer $a < b$, such that ...
57
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6answers
4k 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 ...
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1answer
170 views

Is there a function that grows asymptotically faster than the Busy Beaver numbers?

Is there a function that grows asymptotically faster than the Busy Beaver numbers? That is, I know that BB(n)^n grows faster than ...