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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3answers
269 views

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

Is there a general formula to calculate the n-th digit of any big number?
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2answers
165 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 ...
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1answer
121 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 ...
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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 ...
1
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4answers
79 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
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1answer
474 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
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1answer
243 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 ...
24
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6answers
659 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 ...
30
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1answer
627 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 ...
3
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0answers
477 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
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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 ...
0
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2answers
103 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
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1answer
158 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???
3
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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!!!
2
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1answer
707 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 ...
0
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1answer
151 views

Mix of Modulus and Division

While solving problems in SPOJ, I faced cases where I need to take Modulus of Big numbers like Fibonacci with 10^9 + 7 ( say MOD ). Now, consider the following case : (Fib(n) + Fib(6*n-1)) / ...
4
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1answer
117 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 ...
3
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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 ...
0
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1answer
169 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 $1000!$
13
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1answer
935 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
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1answer
229 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 ...
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 ...
2
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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.
4
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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. ...
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
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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 ...
29
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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) ...
2
votes
2answers
172 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 ...
5
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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, ...
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: $\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$ ...
6
votes
4answers
6k 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 ...
3
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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 ...
62
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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 ...
8
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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.