Skip to main content

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.

Filter by
Sorted by
Tagged with
4 votes
1 answer
128 views

Is there a known generalized Fermat prime with exponent at least $2^{21}$?

In OEIS , the smallest even positive integers $k$ such that $k^{2^n}+1$ is prime are given upto $n=20$. Is there a known prime number with $n\ge 21$ ? Such a number would be very large , so I guess ...
Peter's user avatar
  • 85k
0 votes
1 answer
48 views

Unsolved problems in the relative size of large finite integers

There are a variety of ways to define large numbers (https://en.wikipedia.org/wiki/Large_numbers), such as Graham's number, TREE(3), Rayo's number, etc. Often times we know the relative size of these ...
Jeremy Salwen's user avatar
6 votes
2 answers
141 views

Is $\text{BRANCH}(n)$ finite for $n > 2$?

Is $\text{BRANCH}(n)$ finite for $n > 2$? Define $\text{BRANCH}(n)$ as the maximum length of a string that is composed of at most $n$ unique characters AND meets the following condition: Define a ...
Yash Jain's user avatar
  • 155
2 votes
1 answer
133 views

How big do hyper-reals get?

Let's assume there is some non-standard model of the reals containing a number $N$ that is larger than any real number. Suppose $\exists N\in {^*}\mathbb{R} ( \forall r\in\mathbb{R}: r<N).$ Now I ...
Numeral's user avatar
  • 1,858
0 votes
0 answers
40 views

Which smooth, real-argument fast growing functions are known?

Related to N-ation of N by N I would like to know of a function SFG(x) with following properties: Grows at least as fast as Ackermann sequence Defined for all ...
alamar's user avatar
  • 131
1 vote
1 answer
73 views

Does this text based Tree Function stay finite?

Define $f(k)$ to be the maximal natural number $n$ such that there exist $n$ strings $s_1,\dots,s_n$ from the alphabet $\{1,2\}$ such that no letter is repeated more than $3$ times in a row, for all $...
look at me's user avatar
0 votes
0 answers
78 views

Big Number Textbook recommendation

i want to learn big numbers in 2024, big numbers such as : TREE(3), graham's number, busy beaver, rayo's number. can you guys recommend me some book about big numbers?
JustAsk123's user avatar
0 votes
1 answer
63 views

Does this "tree like" function stay finite?

Or more importantly, does it have a name? The function f(k)=x I have found goes under these rules, There is a group of n strings containing k characters Each string is 1 character longer than the ...
look at me's user avatar
1 vote
1 answer
121 views

Is it possible to calculate the first digits of this number?

Is it possible to calculate the leading decimal digits of $3 \uparrow\uparrow 5\ = 3^{3^{3^{3^3}}} = 3^{3^{7,625,597,484,987}}$? Using currently known methods, this would require knowing the complete ...
Allam A.'s user avatar
  • 229
2 votes
1 answer
93 views

How to get rid of large numbers in a function?

I study mathematics, and I have a question: I have this math function: f(x) = floor(x^99999 / 10^(floor(log10(x^99999+0.1)) + 1 - 5)) (floor() is rounding down) ...
Rick Li's user avatar
  • 21
0 votes
0 answers
99 views

How to approximate Graham's number using $10$ and some notation of large numbers.

Graham's number in Expansion notation approximately equals $3\{\{1\}\}65$ (this is based on $3$). (This notation is explained in https://googology.fandom.com/wiki/Expansion). I want approximation to ...
Mahmoud albahar's user avatar
3 votes
0 answers
106 views

Is there a better way to compute first digits of very large numbers?

The currently known method of finding the first digits of $a^b$ is multiplying $\log_{10} a$ by b, and extracting the fractional part. This allows us to compute the first digits of quite large numbers ...
Allam A.'s user avatar
  • 229
3 votes
0 answers
95 views

In a fast-growing hierarchy, what conditions on $(\alpha,\beta,n)$ imply $f_\alpha(n)<f_\beta(n)$?

Consider any specific fast-growing hierarchy $(f_\alpha)$ defined with one of the commonly used systems of fundamental sequences (e.g., any of those mentioned at the given link, using either Cantor ...
r.e.s.'s user avatar
  • 15k
0 votes
1 answer
133 views

Survival of TREE(3) under iterated logarithms

Instead of base 10 or base e, let the base of our logarithm be a positive integer B. Take log base B of TREE(3), B times, or stop if the answer is negative before B iterations. Say that TREE(3) does ...
Richard Peterson's user avatar
3 votes
0 answers
113 views

Solution for fractional operation "N-ation of N by N"

A lot of articles seems to be concerned with "unimaginably" large numbers, leading to a conclusion that we just do not have sufficiently good notation for dealing with such numbers. So ...
alamar's user avatar
  • 131
7 votes
1 answer
423 views

TREE(3) and the Goodstein sequence

TREE(3) is an extremely large number that requires ordinal arithmetic to prove it is finite. For what value of n would $G(n)>TREE(3)$? The length of the Goodstein sequence $G(n)$ is how many ...
Sheldon L's user avatar
  • 4,544
0 votes
1 answer
152 views

Where is $G64\uparrow \uparrow G64$ in the Fast-growing hierarchy?

I am trying to find out how $G64\uparrow \uparrow G64$ can be represented by the Fast-growing hierarchy, but I do not know how this can be done. Is there a way to simply convert between those two ...
Abanob Ebrahim's user avatar
1 vote
1 answer
40 views

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 ...
Peter's user avatar
  • 85k
5 votes
0 answers
95 views

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 ...
Peter's user avatar
  • 85k
3 votes
0 answers
248 views

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^{...
Peter's user avatar
  • 85k
2 votes
0 answers
90 views

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) , ...
Peter's user avatar
  • 85k
1 vote
0 answers
96 views

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 ...
Mahmoud albahar's user avatar
4 votes
1 answer
177 views

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 ...
Peter's user avatar
  • 85k
4 votes
0 answers
165 views

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&...
Peter's user avatar
  • 85k
0 votes
1 answer
49 views

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 ...
ayr's user avatar
  • 731
2 votes
1 answer
389 views

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 ...
CPlus's user avatar
  • 183
0 votes
0 answers
399 views

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 ...
MathStackexchangeIsMarvellous's user avatar
1 vote
0 answers
156 views

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 ...
Hovercouch's user avatar
  • 2,688
3 votes
1 answer
158 views

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 ...
Isaac King's user avatar
4 votes
0 answers
105 views

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 ...
Hubert Schölnast's user avatar
8 votes
5 answers
598 views

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 ...
Hubert Schölnast's user avatar
3 votes
1 answer
235 views

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. ...
Jumper_Child64's user avatar
2 votes
0 answers
116 views

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 $n>2$ ? ...
Peter's user avatar
  • 85k
4 votes
1 answer
225 views

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 $600$ ...
mr_e_man's user avatar
  • 5,706
0 votes
0 answers
128 views

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 ...
esi's user avatar
  • 11
1 vote
1 answer
71 views

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 ...
Jacob.C's user avatar
  • 13
1 vote
0 answers
449 views

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} ...
Nikoloz Chichua's user avatar
-1 votes
1 answer
186 views

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 ...
skoestlmeier's user avatar
1 vote
1 answer
361 views

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?
C Shreve's user avatar
  • 571
1 vote
1 answer
92 views

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 ...
entropy's user avatar
  • 107
0 votes
0 answers
111 views

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 ...
Xavier Prudent's user avatar
0 votes
1 answer
126 views

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 $&...
Peter's user avatar
  • 85k
3 votes
1 answer
184 views

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 ...
Peter's user avatar
  • 85k
0 votes
0 answers
151 views

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, ...
0xff's user avatar
  • 121
1 vote
2 answers
160 views

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 ...
Nicholas Hernandez's user avatar
1 vote
3 answers
181 views

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}+...
Mike Pierce's user avatar
0 votes
0 answers
81 views

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 ...
xycf7's user avatar
  • 131
3 votes
1 answer
381 views

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^...
Polv's user avatar
  • 149
5 votes
0 answers
394 views

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 ...
PanJanek's user avatar
  • 169
1 vote
0 answers
51 views

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 ...
Moe Sanjaq's user avatar

1
2 3 4 5 6