Questions tagged [big-numbers]

For questions relating to the computation, estimation and properties of extremely large finite quantities that are not usually used in mainstream mathematics. This is not for questions that just have large numbers; the fact that a number is very large has to affect the question.

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Implementing binary splitting

I found out this article which says that binary splitting can be used to compute operations on numbers with high precision (sort of BigNum), such as exponential, trigonometric functions and constants (...
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Calculate two to the power of decimal

I have a decimal number $a$ with integer and fractional parts as follows: $$ a_{int} = a_0 + a_1 2 + a_2 2^2 + \cdots $$ $$ a_{fr} = \frac{a_{-1}}{2} + \frac{a_{-2}}{2^2} + \cdots $$ I want to ...
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How to understand the fast divisibility check for a double-width dividend?

I found this algorithm from the GNU factor utility. Given a double width dividend $n=n_1B + n_0$ and a single width odd divisor $d$, where $n_1, n_0, d < B=2^w, 2\nmid d$. Then with the precomputed ...
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Is this number, N, greater than Graham’s Number?

So, using Knuth’s up-arrow notation, if 3 ↑ 3 = 3^3 and 3 ↑ ↑ 3 = 3 ↑ (3 ↑ 3) = 3 ↑ 27 = 3^27 Then consider number N defined by Knuth's up-arrow notation: $$N = googolplex\uparrow^{googolplex} ...
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Calculate the n-th number of a power with huge exponent

There are several questions asked (e.g. 1, 2 or 3) on the last digit of numbers like $7^{355}$ or $237^{222222212202237}$. My question is, if there is any efficient method to calculate the n-th digit ...
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Is the number of digits in Graham's number greater than the number of protons in universe (~10^122)?

I am wondering, is the number of digits in Graham's number greater than the number of protons in the known universe (~10^122)? Or is there some other 'big' lower bound to the number of digits in G64?
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Probability of a big number that is composed of two special numbers to be multiple of 19

A big number, which is 117179 digits in length, was constructed by concatenation of only two special numbers, which are 40 and 8 (both 40 and 8 must be used). There are around 62870 of any of those ...
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How to convert an integer to a Knuth notation?

The Knuth notation enables to write very large number in a compact way. I however cannot find the way to do the reverse calculation.For instance, given the number $123456789$ , how can it be written ...
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Growth rate of Conway-chains with very large first entry?

The rules of the Conway chained arrow notation are here Some approximations in the fast growing hierarchy are given for various Conway-chains. But what about the growth rate, if the first or the first ...
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Where in the fast growing hierarchy is the next level beyond Latri?

I found the huge number Latri here If I understand it right, it can be written as $< 3 , 3 , 3 >$ $< 2 >$ with Bowers 2-dimensional arrays. Is this correct ? I wonder where the number $&...
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Lower bound for $bb(7)$ - which one is true ? And what is the bound for $bb(6,3)$?

$bb(7)$ is already extremely large , but I found a discrepancy in the lower bound: In this survey the lower bound for $bb(7)$ is given as $$ BB(7) > 10^{10^{10^{10^7}}}, $$ hence four tens in the ...
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Converting power of 10 to Knuth Arrow notation

I have this really big number $10^{1762613844998129336721604609} - 1$ which I want to represent in a more compact way. I know it's possible to convert these kind of numbers to Knuth arrow notation, ...
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What is (319!)!?

This question came to my mind when someone wrote 319!! on my dorm (room 319). I was able to find 319! and it's about 10^661. I used mathematica to compute (n!)! and I was able to go up to 12 and came ...
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Evaluating $\frac{100x^{100}+98x^{98}+96x^{96}+\cdots+6x^6+4x^4+2x^2}{99x^{99}+97x^{97}+95x^{95}+\cdots+5x^5+3x^3+x}$ for $x=99$ billion

Approximately what is the value of $f$ evaluated at $99$ billion, where \begin{equation*} f(x) = \frac{100x^{100} + 98x^{98} + 96x^{96} + \dotsb + 6x^6+4x^4+2x^2} {99x^{99}+97x^{97}+...
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Theorems only valid for huge numbers [duplicate]

I am trying to find a theorem that is valid only for very large numbers. Example: There are numbers which have more than 100 distinct factors. Above theorem satisfies this condition, but it is a ...
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How do I calculate the first n digits and last m digits of $3^{3^{3^{3^{3}}}}$?

It is possible with $3^{3^{3^{3}}}$, from this algorithm (https://stackoverflow.com/questions/68797298/calculating-3333-very-large-exponent-how-did-wolfram-do-it). However, being a large number, $3^{3^...
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Random search for very big Collatz conjecture counter-examples

I know that exhaustive search was done to test numbers up to 2^68. This seems like a big number but when looking at Collatz function as a Turing machine manipulating some input bit sequence, only ...
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Birthday problem but with $2^{128}$ different days in the year [duplicate]

I am trying to calculate how many randomly generated ids I need to produce for there to be a 1% probability I get a duplicate id. There are $2^{128}$ possible ids. I understand this is just the ...
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4 votes
3 answers
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Examples of rational numbers with large denominators appearing unexpectedly. [duplicate]

I am looking for examples of rational numbers with large denominators that pop up in questions in which there is a-priori no large numbers involved or no obvious reason why the denominator would be ...
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Most efficient way to square modulo a Mersenne prime

I realise this question is somewhere in between Math StackExchange and StackOverflow. So forgive me if this is too much of a practical question, it would probably too theoretical elsewhere. I am ...
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What is the equivalent of breaking a n bits key with “winnig lottery every minute x times in a row”?

I was wondering how to give a hint to someone to imagine the size of a 256 bits key speaking about Bitcoin (even if private key in Bitcoin are 160 bits if I remember) and not with the number of atoms ...
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What is the largest finite number you can make using no more than 6 characters?

(I am new to the Math Stack Exchange community so tell me if this question is not allowed, however, I checked on Meta first.) (Also, I don't really know which tags I should assign this question, so ...
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A confusion about the TREE function.

I know, from Kruskal's tree theorem, that the sequence of trees mentioned in the TREE function cannot be infinite. However, why can't the sequence of trees get arbitarily large, so that there is no ...
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When does Busy Beaver surpass TREE(3)?

I saw a question which asked, "When does Busy Beaver surpass TREE(n)?" I am asking a somewhat different question, about a specific value of TREE. I know that TREE(3) is an unimaginably vast ...
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First value of the second order busy beaver function?

There is a bit of talk about the busy beaver function. It was asked whether or not it is possible to make a function that grows faster than the normal busy beaver function. One way to do this is to ...
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1 vote
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A confusion regarding Rayo's number and Busy Beaver function.

I am slightly confused about the definition of the Rayo function and Rayo's number, and how it relates to the Busy Beaver function. I know that ZFC can't pin down the precise value of even $BB(7918)$. ...
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Linearizing the product of a binary and a continuous variable

I have an MIP optimization problem that has a constraint $p\geq xy$, where $x$ is a binary variable, $p$ and $y$ are non-negative continuous variables. I tried the Big-M method. However, the upper ...
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prime number (a form like Mersenne primes)

I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a ...
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5 votes
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Comparing power towers of $2$s and $3s$

Let $x=[x_1,x_2,...,x_n]$ be a finite list of positive real numbers, and define $\tau x$ as the power tower formed by these numbers. The function $\tau$ can be recursively defined by the following two ...
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2 votes
1 answer
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What is the chance that a number $P$ is prime if it's not divisible by any number less than $x$?

I am trying to check if a very big number ($>10^{10,000,000}$) is possibly prime. I have written a computer program to check if the number has any smallish (less than like $600,000,000$) factors......
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calculating Modulus on Massive numbers [duplicate]

In the case I have two numbers large enough to justify using scientific notation twice $A \times 10^{B \times 10^C}$ or $Ae+Be+C$ How would I calculate Modulo without taking the numbers or any part of ...
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Given that $2017$ is prime, how do I prove this statement?

I'm asked to prove the following statement: Let $N=(1008!)^2+1$. Prove that $N$ is divisible by $2017$. (Hint: $2017$ is prime.) I don't know how to go about proving this statement, since there seems ...
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How to get the last $n$ digits of Ackermann function?

The Ackerman function is defined as follows: $$A(m,n)= \begin{cases} n+1,& m= 1\\ A(m-1,1), & m>0, n=0\\ A(m-1, A(m,n-1)), &m,n>0 \end{cases}$$ Is it possible to get the last ...
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Is there a property relating gcd(a, b) % c and gcd(a %c, b%c) % c. Here all a,b,c are natural numbers i.e.>0 and '%' represents modulo operator.

I have been facing some difficulty solving problems relating the gcd and modulo operator. If I use the Euclidean algorithm then it works fine when either a or b is small because in one step itself a ...
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2 answers
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How big are $f_3(3)$ and $f_4(4)$ in the fast growing hierarchy?

I read some articles about the fast growing hierarchy (and saw some vids), and I wonder how to calculate: $f_3(3)$ And especially $f_4(4)$ I know that there are huge numbers, but I wonder if someone ...
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Will this function grow faster than Busy Beavers as $n \to \infty$

Consider the following function: $$f(x)=x \uparrow ^{x} x$$ Where the notation $\uparrow$ is Knuth's up-arrow notation and $\uparrow ^{n}$ means $n$ number of up-arrows. For example, $2\uparrow ^{4}...
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How many ways can you set the table for 2 people with n distinct cups and r distinct plates?

This problem came up at breakfast. Alice and Bob have 20 different coloured cups and $10$ different coloured plates. They always set the table so that each has a plate and a cup. How many different ...
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1 vote
1 answer
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In what sense would Graham's tree be virtually the same as TREE(3)?

How Big would "Graham's Tree" be? I'm coming off this post which asks about the size of the number which we would obtain by replacing the 3's in Graham's number construction by TREE(3). ...
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is it possible to calculate mod of a very long number manually? [closed]

Assuming we found a very old book with more than 1000 years old and there is an extraordinary mathematical pattern in it. Let's say its text has 'many' clear patterns. As an example, concatenation of ...
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Do formulas use unary functions in first-order oodle theory?

BIG FOOT is supposedly a finite number which is defined using first-order oodle theory (an extension of set-theory) like defined here. It has been shown here that it's not well-defined. The reasoning ...
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Comparing large exponents.

I have come up with a way to compare large exponents, for example: I can tell which number is bigger in $12345^{78901}$ or $21346^{78900}$ within a few seconds without using calculator. So I have 2 ...
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Big number short representation / approximation

What's an efficient way of representing a big number (up to 100M digits) in a short format. I'm thinking on possible solutions: logs factoriadic prime numbers base Would like precision > 80-90% ...
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What information about the factors of a big number can be gained in a couple of minutes?

Given a number with some hundreds of decimal figures, it's possible to check if it is a prime number if there are small factors if there are factors close to the square root. But what else? Are ...
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There exists infinitely many primes formed of 2^(F[n!+1])-1

I have a conjecture: Ultra-Primes-Conjecture. There exists infinitely many primes formed of $2^{F[n!+1]}-1$. Here $2^p-1$ is Mersenne number, $F[n+2]=F[n+1]+F[n], 0,1,1,2,3,5,8,...$ is Fibonacci ...
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Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is ...
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Can I avoid the overflow which I get with the digits of $f_3(3)$?

I would like to ananlyze the digits of the number $$f_3(3)$$ which is defined as follows : Start with $\ n=3\ $ and apply the opration $\ n\cdot 2^n\ $ three times. The first iteration gives $\ 24\ $,...
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2 votes
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Rightmost decimal digits of Graham's number

How to find rightmost $n$ decimal digits of Graham's number efficiently. The last 500 digits are on the wiki/Graham's_number, but I want to know more. PowerTowerMod seems to be able to do it but is ...
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Can someone explain TREE(3) in extremely simple terms?

I have recently begun getting interested in the field of googology, or as your tags list it, "big numbers." One of the first things I saw mentioned was the famous TREE(3) function. I am at a high ...
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Shannon's number growth rate [closed]

Where does Shannon's number lie on the fast growing hierarchy? Also, consider the function of (# of moves so far) -> (# of chess games). How does the growth rate of this function compare to the ...
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1 vote
1 answer
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A question about relation between two fast growing sequences.

Let $TREE(n)$ and $tree(n)$ are Kruskal' tree sequences. The second one is called weak. Prove that $TREE(3)>tree^{tree^{tree^{tree^{tree^{8}(7)}(7)}(7)}(7)}(7)$ You can see that inequality in ...
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