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

learn more… | top users | synonyms

62
votes
5answers
5k 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 ...
30
votes
1answer
628 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
29
votes
3answers
1k 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) ...
24
votes
6answers
660 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 ...
13
votes
1answer
937 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 !
8
votes
5answers
442 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 ...
8
votes
4answers
9k 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.
8
votes
2answers
562 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. ...
7
votes
2answers
313 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 ...
7
votes
1answer
2k 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
votes
4answers
7k 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 ...
6
votes
1answer
345 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 ...
5
votes
3answers
2k views

Estimating a certain row of Pascal's triangle

I need to calculate all the numbers in a certain row of Pascal's triangle. Obviously, this is easy to do with combinatorics. However, what do you do when you need to estimate all the numbers in, say, ...
5
votes
1answer
1k 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 ...
5
votes
2answers
146 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
118 views

Are there any secure ciphers you can use without a computer?

I have some kids that like encryption schemes such as the Caesar cipher and the Vigenère cipher. I would like to teach them something that's not easily breakable by todays maths and computers, but I ...
4
votes
2answers
230 views

RSA: Creating a key of desired length

Thanks and with respect to the users of this site, I've succeeded in creating an Encryption/Decryption procedure for the RSA algorithm. I also implemented a Miller-Rabin probabilistic primality test. ...
4
votes
1answer
106 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 $n\ge 3$, ...
4
votes
1answer
24 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, ...
4
votes
0answers
144 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 ...
3
votes
3answers
236 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!!!
3
votes
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 ...
3
votes
2answers
3k views

How to handle big powers on big numbers e.g. $n^{915937897123891}$

I'm struggling with the way to calculate an expression like $n^{915937897123891}$ where $n$ could be really any number between 1 and the power itself. I'm trying to program (C#) this and therefor ...
3
votes
1answer
135 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 ...
3
votes
1answer
340 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 : $$M_1 := 10$$ $$M_2 := 10^{10^{10^{...10}}}$$ ...
3
votes
1answer
4k 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 ...
3
votes
2answers
59 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 ...
3
votes
1answer
332 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
votes
2answers
137 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 ...
3
votes
1answer
590 views

How to solve an inequality containing the sum of factorials and powers

In previous question, I asked how one would simplify the following equation for the case where the variables are very big: $\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$ ...
3
votes
0answers
481 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 ...
2
votes
2answers
745 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 ...
2
votes
1answer
2k views

Calculate the root of a number without useing the root function or decimal numbers

I'm trying to build a program in c# which will calculate prime numbers for me. I'm using the BigInteger class to work with 'endless' numbers. However, there is a big down side on this function, I ...
2
votes
2answers
98 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 ...
2
votes
1answer
171 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 ...
2
votes
1answer
152 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
votes
2answers
24 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? ...
2
votes
2answers
313 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 ...
2
votes
1answer
711 views

How to calculate modulo of large integer (number having 25000 digits)

I'm looking for solution to a problem to calculate modulo of very large number that can contain 25000 digits or less (n) with 10 digit number (m). ( n % m ) ? Pointer to appropriate theory resource ...
2
votes
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.
2
votes
2answers
173 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 ...
2
votes
2answers
73 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 ...
2
votes
1answer
50 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 ...
2
votes
1answer
105 views

Is the estimation of number's name's length and comma-grouping feasible?

I am thinking in a mathematical problem that probably is already formulated and even solved. It is about big integers and someting else. Let n be an integer positive number. For n := 1,000 we have ...
2
votes
0answers
68 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
1answer
24 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: ...
2
votes
0answers
112 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 ...
2
votes
1answer
47 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 ...
2
votes
1answer
41 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 ...
2
votes
0answers
69 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 ...