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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3
votes
0answers
54 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 ...
0
votes
1answer
30 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
50 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
41 views

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

Can log 2^274,207,281,512 in base 512 be calculated?
1
vote
1answer
107 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
1answer
83 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: ...
4
votes
0answers
71 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
29 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 ...
0
votes
1answer
63 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 ...
5
votes
4answers
121 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
93 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
42 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
49 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
189 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: $2^2=4$ $2\uparrow\uparrow2=216$ ...
3
votes
1answer
83 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 ...
0
votes
1answer
75 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 ...
0
votes
0answers
50 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
votes
0answers
92 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
30 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
68 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
67 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 ...
2
votes
2answers
95 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
1k 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
39 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 ...
8
votes
5answers
571 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
29 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
46 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
44 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 ...
3
votes
3answers
232 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
0answers
175 views

Node potentials of minimum cost flow successive shortest path 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 ...
0
votes
2answers
236 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 ...
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
81 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
234 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 ...
1
vote
2answers
77 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
105 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
160 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 ...
2
votes
1answer
55 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 ...
4
votes
1answer
116 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 $n\ge 3$, ...
1
vote
1answer
165 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 ...
2
votes
1answer
45 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 ...
1
vote
1answer
96 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 ...
1
vote
1answer
110 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 ...
3
votes
2answers
823 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 ...
5
votes
2answers
153 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 ...
4
votes
1answer
28 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, ...
6
votes
1answer
412 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 ...