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
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0answers
53 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 ...
24
votes
7answers
766 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 ...
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
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: $2^2=4$ $2\uparrow\uparrow2=216$ ...
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 ...
14
votes
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 !
1
vote
1answer
106 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?
-3
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2answers
41 views
69
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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 ...
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: ...
6
votes
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
votes
0answers
69 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 ...
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 ...
2
votes
0answers
41 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 ...
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 ...
33
votes
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
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) ...
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 ...
1
vote
1answer
108 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
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
91 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? ...
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 ...
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
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 ...
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
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 ...
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 ...
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 ...
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} ...
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
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 ...
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 ...
2
votes
1answer
5k 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 ...
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
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
1answer
87 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 ...
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 ...
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 ...
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 ...