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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14
votes
1answer
713 views

How many digits of the googol-th prime can we calculate (or were calculated)?

Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\...
0
votes
3answers
96 views

What is the Googol root of a Googolplex? [closed]

$\text{Googol}=10^{100}$ $\text{Googolplex}=10^{\text{Googol}}=10^{{10}^{100}}$ What is $\sqrt[\text{Googol}]{\text{Googolplex}}$? I know that's the same as $\sqrt[10^{100}]{10^{10^{100}}}$ but I ...
1
vote
2answers
86 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 $...
0
votes
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
votes
1answer
57 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
votes
1answer
70 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 ...
2
votes
1answer
81 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
votes
2answers
147 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 ...
0
votes
1answer
110 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 \...
3
votes
3answers
109 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 $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
2
votes
1answer
56 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 ? ...
3
votes
0answers
91 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
votes
1answer
50 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 ...
9
votes
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
vote
2answers
61 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 ...
-3
votes
2answers
44 views

Calculating log 2274,207,281,512 in base 10. [closed]

Can log 2^274,207,281,512 in base 512 be calculated?
2
votes
1answer
123 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?
2
votes
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\...
4
votes
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
votes
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
votes
1answer
71 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 ...
6
votes
4answers
149 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 $G$ ...
1
vote
2answers
99 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 ...
2
votes
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 ...
4
votes
2answers
206 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) ...
-1
votes
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=...
3
votes
1answer
85 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
89 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......
0
votes
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 $m,n>1$ ...
1
vote
0answers
94 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
32 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? ...
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
votes
1answer
83 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}...
2
votes
2answers
98 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
votes
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
vote
1answer
43 views

Comparison of arbitary conway chains (in particular a chain with $m$ $m's$) to $f_{\omega^2}(n)$

Wikipedia describes the Conway chained arrow notation and the fast growing hierarchy. I learnt that the function $f(n):=\large f_{\omega^2}(n)$ has the same growth rate as the function $g(n):=n\...
8
votes
5answers
588 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 ...
2
votes
1answer
30 views

find least multiple formed only of 1's of given number

The problem states that given a number find the least multiple formed only of 1's. If no such number exists then 0 will be the answer. For example for: ...
0
votes
4answers
50 views

How to manually determine big number congruences

How is it possible to determine if the the following congruence is true manually, with resort to a basic calculator? The real problem here is how to do the math with a such big number? $$ 2015^{50} \...
1
vote
1answer
57 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\...
3
votes
3answers
249 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 \...
0
votes
2answers
247 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&...
0
votes
0answers
28 views

Check for greatest common multiple in a range

I am supposed to be writing a program that takes two numbers and checks to see if they have a common multiple in the range of 1 - 1100000000, 1.1 billion. If there are many common multiples, I have to ...
1
vote
2answers
86 views

Which arrangement produces the largest number?

I learnt that the power tower $2\uparrow3\uparrow4\uparrow...\uparrow n$ is larger than any power tower with a different order of the numbers $2,3,4,...,n$. Is this also true for conway-chains and ...
0
votes
3answers
250 views

Modulus calculation for big numbers

I am having problems with calculating $$x \mod m$$ with $$x = 2^{\displaystyle2^{100,000,000}},\qquad m = 1,500,000,000$$ I already found posts like this one https://stackoverflow.com/questions/...
1
vote
2answers
87 views

What are sharp lower and upper bounds of the fast growing hierarachy?

With fast growing hierarchy, I mean the Wainer hierarchy, which starts with $$f_0(n)=n+1$$ $$f_1(n)=2n$$ $$f_2(n)=2^n n$$ A lower bound for $f_m(n)$ is $2 \uparrow^{m-1} n$. If $f(n,m):=2 \...
1
vote
1answer
113 views

Fast growing hierarchy : How can I show that any sequence grows faster than the one before?

How can I show, that in the fast growing hierarchy, every sequence grows faster than the one before ? A function $f(n)$ is said to grow faster than a function $g(n)$, if for every $k$ there exists ...
2
votes
0answers
173 views

subtract a number from its digits until it reaches 0 [closed]

Can anyone help me with some algorithm for this problem? We have a big number (19 digits) and, in a loop, we subtract one of the digits of that number from the number itself. We continue to do this ...
0
votes
0answers
71 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 ...