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.
253
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Largest known prime maximizing the relative digitsum?
A prime number with a given number of digits (except $1$) has the maximum possible digitsum (in base $10$) , if it has only digits nine except one digit which is $8$.
A large (probable) prime of this ...
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Largest known prime with digit sum $4$?
According to OEIS , the number $$10^{509546}+3$$ is a (I guess probable) prime with digit sum $4$ in base $10$.
Is it the largest known such prime ?
What is the largest known such prime if it must ...
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Is the quadratic residue the only restriction?
I still search without success for a prime factor of the huge number $$2\uparrow \uparrow 4+3\uparrow \uparrow 4$$
Another way to write this is $$3^{3^{3^3}}+2^{2^{2^2}}=3^{3^{27}}+65536=3^{...
- 82.2k
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Is there any algorithm better than trial division to factor huge numbers?
Suppose , we want to find prime factors of a huge number $N$ , say $N=3^{3^{3^3}}+2$. We can assume that we can find easily $N\mod p$ for some positive integer $p$ (as it is the case in the example) , ...
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Is it possible to calculate the size of the sum of the first Graham's number terms of the harmonic series?
I am curious about the size of the sum of the first Graham's number terms of the harmonic series. The harmonic series is a well-known mathematical series, and Graham's number is an incredibly large ...
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72
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Is the "reverse" of the $33$ rd Fermat number composite?
If we write down the digits of the $33$ rd Fermat number $$F_{33}=2^{2^{33}}+1$$ in base $10$ in reverse order , the resulting number should , considering its magnitude , be composite.
But can we ...
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Best known lower bound for $bb(7)$ after Pavel Kropitz's breakthrough?
Concerning the busy beaver with $6$ states , there recently occured a major breakthrough : Pavel Kropitz has proven $bb(6)>10\uparrow \uparrow 15$.
The old record was just in an "astronomical&...
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from p-1 algorithm in FP to design a p+1 algorithm in FP^2
the (p-1) algorithm is based on the properties of the field of the finite field
𝐹𝑃, using instead the field 𝐹𝑃 2̅̅, develop a (p+1)-factoring algorithm for
use when a private P factor of N is such ...
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Mapping a vector to a unique number using statistical coefficients and vice versa
Working with sequences of various dimensions of random integers, I saw that Wolfram Alpha provides information on a number of statistical indicators for this sequence.
Among these indicators, for ...
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Estimation of distibution entropy based on a large sample
I'm looking for an efficient way to estimate entropy of continuous distribution based on a large (millions of samples) sample drawn from it. I've found a few papers (example: https://www.sciencedirect....
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How do we know that Loader is bigger or grows faster than TREE or SSCG?
From what I have gathered online about these numbers, they say that Loader's Number is larger than TREE(3) or SSCG(3) or similar. The reasoning I have seen goes is that Loader's Number is the largest ...
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Which is the largest number out of these three$?$
Rayo's Number
The smallest number bigger than any finite number named by an expression in the language of first-order set theory with a googol symbols or less.
Now everyone knows that it is the ...
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Fastest growing continuous functions?
In Who can name the bigger number?, Scott Aaronson gives two examples of fast-growing functions:
The Ackermann sequence, defined specifically as ...
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1
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Which of these two very large numbers is larger?
I made the horrible mistake of hosting a Scott Aaronson style "large numbers" competition. (I used a character limit to prevent it from getting too crazy.)
I was able to knock out most of ...
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Why is $\mathrm{gcd}(n^a+b,(n+1)^a+b)$ almost always prime when $n$ is large?
Given is this function:
$$
m=\mathrm{gcd}(n^a+b,(n+1)^a+b)\\
a,b,c\in\mathbb{N}
$$
On OEIS A118119, for $2\leq a\leq 84$ and $b=1$, the smallest values for $n$ in each case where this expression is ...
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5
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Algorithm that runs very long for some parameters and never halts for others
11 days ago I asked a similar question with a specific narrow focus, to which there is probably no answer. So here I am asking about the underlying problem.
I teach computer science and would like to ...
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3
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1
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How was Little Graham's number calculated?
I’ve been interested in Graham’s number for a while now however I've been unable to figure out how little graham was derived in the paper “RAMSEY'S THEOREM FOR n-PARAMETER SETS” BY R. L. GRAHAM AND B. ...
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Some questions about the prime factors of a very fast growing function
Let $$f(n)=\sum_{j=1}^n (j!)^{j!}$$ for integer $n\ge 1$
The first few values are $f(1)=1 , f(2)=5 , f(3)=46661 , f(4)=1333735776850284124449081472890437$
Is $f(n)$ a prime number for any $n>2$ ?
...
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How many sets of non-adjacent edges in the 600-cell?
(This is a simpler variant of my previous question, with one colour instead of two.)
The 600-cell is a 4D regular polytope. It has $120$ vertices, $720$ edges, $1200$ triangular faces, and $600$ ...
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Calculate two to the power of decimal
I have a decimal number $a$ with integer and fractional parts as follows:
$$ 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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1
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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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0
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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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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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122
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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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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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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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2
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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 ...
5
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2
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174
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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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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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99
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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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