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52
votes
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 ...
29
votes
1answer
548 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 ...
23
votes
5answers
460 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 ...
7
votes
4answers
5k 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.
6
votes
2answers
160 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 ...
5
votes
3answers
3k 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 ...
4
votes
3answers
1k 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, ...
4
votes
1answer
107 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
225 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
420 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 ...
4
votes
1answer
2k 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
3answers
225 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
1k 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
110 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
551 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
65 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
0answers
260 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
0answers
305 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
1answer
1k 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
88 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
141 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
510 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
1k 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
170 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
1answer
49 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
votes
2answers
252 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
100 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
70 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 ...
2
votes
0answers
60 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 ...
1
vote
3answers
138 views

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

Is there a general formula to calculate the n-th digit of any big number?
1
vote
2answers
97 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 ...
1
vote
1answer
105 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 ...
1
vote
1answer
110 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 ...
1
vote
4answers
74 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
vote
1answer
126 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 ...
1
vote
1answer
147 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 ...
1
vote
1answer
196 views

Normalize only big numbers for plotting

I have a set of numbers: [9, 8, 6, 4000] I want to plot a bar chart and I want to normalize only the 4000 number to 4, so the range of Y axis will be [0, 9]. Under the 4 bar I would write * 1000 so ...
1
vote
0answers
70 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 ...
0
votes
1answer
129 views

How to compare big numbers that are outcome of different functions.

How is the best way to compare big numbers? They are result of two functions with different asymptotic growth. For example: Googleplex which is $10^{{10}^{100}}$ to $1000!$
0
votes
1answer
112 views

Ackerman numbers, arrow notation

How to compare 3^3^3^3 and to 3↑(3↑↑3). (Ackerman number, arrow notation) Are these two numbers equal??? How to compare 3↑(3↑↑3) with googol and googolplex???
0
votes
1answer
29 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.
0
votes
2answers
79 views

What is the biggest known safe prime number?

I am looking for the biggest known safe prime number. Can someone provide some reference to what that number is and a proof that it is indeed a safe prime number?
0
votes
1answer
77 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 ...
0
votes
0answers
37 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 ...
0
votes
1answer
31 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 ...
0
votes
0answers
45 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
votes
0answers
24 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 ...
0
votes
0answers
35 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 ...
0
votes
0answers
49 views

Tight bounds for the number “2 in a hexagon” wanted (Steinhaus-Moser-Notation)

The Steinhaus-Moser-function is defined in the following way : $$M(n,1,3) = n^n$$ $$M(n,1,p+1) = M(n,n,p)$$ for all $p\ge3$ $$M(n,m+1,p) = M(M(n,1,p),m,p)$$ for all $p\ge3$ and $m\ge1$ The ...