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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.

-1
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2answers
37 views

A Question on Graham’s number [on hold]

I have read an article that stated this: “I can’t tell you the number of digits it has. I can’t tell you how many digits the number of digits it has. I can’t tell you how many digits the number of ...
2
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1answer
44 views

An analogue to Knuth's up-arrows

For every prime $p$, $p\#$ is defined as $p\#=2\cdot 3\cdot 5\cdots p$, let us further define $p\#\#=2\#\cdot 3\#\cdot 5\#\cdots p\#$, $p\#\#\#=2\#\#\cdot 3\#\#\cdot 5\#\#\cdots p\#\#$ and so on. ...
3
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2answers
67 views

Compare different base powers-towers (of 'height' five)?

Let's say I want to compare two numbers that are stacked powers of different bases: $a^{b^{c^{d^e}}}$ compared to $f^{g^{h^{i^j}}}$ where all ten values will be integers in the range $[1,10]$. ...
0
votes
1answer
38 views

big number remainder [duplicate]

$333^{222}$ mod $7$ Modulo 7: $333^1 = 4$ $333^2 = 4*4 = 2$ $333^3 = 2*4 = 1$ $333^4 = 1*4 = 4$ And so on... we already got the remainder pattern of the powers of $333$ over $7.$ The remainder ...
0
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2answers
51 views

Inequalities between large numbers?

It's been shown Gaham's Number g₆₄ is way larger than Moser's Number (< g₃), itself larger than Skewes' Number {≈(10↑↑4)34}. How about the position of Grahal g₁ = 3↑↑↑↑3 (or Triteto) with respect ...
0
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1answer
45 views

What is the value of X in (3,3(1)X,2) = (3,3(1)3,3)

I was reviewing Deedlit's awesome explanation for how the rules of planar arrays work at How can the number $\left\langle \matrix {3&3\\3&3}\right\rangle $ be described? as well as https://...
0
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1answer
117 views

What is the language of FOST (First Order Set Theory)

I’ve been reading about Rayo’s number and I’m finding it difficult to grasp what exactly the language of FOST is. I understand the concept of finding the smallest finite number greater than any ...
2
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4answers
90 views

How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?

3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms ...
1
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1answer
275 views

What is explanation of $\text{Sat}([φ],s)$ in the definition of Rayo’s number?

The definition of $\text{Sat}([φ],s)$ can be found here. All I want is an explanation of what each line in this definition means and how $\text{Sat}([φ],s)$ works. The only relevant thing that I ...
3
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1answer
121 views

lower bound for Kruskal's weak tree function

The wiki on Kruskal's tree theorum briefly mentions the weak tree function regarding unlabeled trees. It gives values of tree(1) = 2, tree(2) = 5 (trivial to prove) but then it gives tree(3) >= 262140....
3
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4answers
793 views

What is the best algorithm for finding the last digit of an enormous exponent? [duplicate]

I found most answers here not clear enough for my case such as $$ 123155131514315^{4515131323164343214547} $$ I wrote the $n\bmod10$ in Python and execution time ran out. So I need a faster ...
2
votes
1answer
130 views

Question about $TREE(3)$ and Graham's Number

Lets say: $G=\text{Graham's Number}$. And: $$ \begin{align*} \alpha_1&=G\uparrow^G G, \\ \alpha_2&=\alpha_1\uparrow^{\alpha_1} \alpha_1 \\ &\vdots\\ \beta_1 &= \alpha_G\uparrow^{\...
8
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0answers
157 views

What is the largest number used in a useful mathematical proof that isn't just an upper or lower bound? [closed]

There are quite a few famous gigantic numbers used in useful mathematical proofs, like Skewes's number, Graham's number, and the number $2 \uparrow \uparrow 10^{10^6}$ from Coward and Lackenby's 2011 ...
3
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0answers
310 views

When does the busy beaver function surpass TREE(n)? [closed]

Since TREE is a computable function the BB function grows faster than it, but TREE seems to grow much more quickly early on, so when does Busy Beaver surpass it?
13
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4answers
1k views

Sum of digits of sum of digits of sum of digits of $7^{7^{7^7}}$

On the back of a mathematical magazine, I came across some "quick facts" about the number 7. Most of them were real life related ones, like "Rome was buit on 7 hills", "the neck of most mamals is made ...
0
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0answers
61 views

(a / b) mod p for large a and b

i get stuck in finding $(a / b) \bmod p$ what i did is googled and saw some theories and i did $(a/b) \bmod p = ((a \bmod p) * (((b^{\,p - 2} \bmod p)) \bmod p)) \bmod p$ where $p$ is prime number $(...
0
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2answers
126 views

There is a way to write TREE(3) via $F^a(n)$?

I read about Graham number and TREE(3). Graham number is: $f^{64}(4)$ where $f(n)=3\uparrow^n 3$ My question is: If there is a way to write ...
3
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2answers
127 views

Where does this array-based fast-growing function fall in the fast-growing hierarchy, and how does it compare to TREE(n)?

[See below for a clarification edit and progress thus far] So I have been reading into the "fast-growing hierarchy" of functions, and I devised this (somewhat convoluted) function for generating very ...
1
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2answers
92 views

Is there any function that like this function?

I got a idea from fast-growing hierarchy function to create new function g.(I think it is computable.) $$g_0(n) = n + 1$$ $$g_{a+1}(n) = g_a^{g_a(n)}(n)$$ Which different from fast-growing ...
7
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1answer
157 views

Is it true that $\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$ has period $18$ and can never take the values $3$ and $6$?

Is it true that the arithmetical function $f:\mathbb{N}\setminus\{1\}\rightarrow \mathbb{Z}_9$ given by $$f(x\mid k)=\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$$ has period $...
3
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1answer
158 views

Super-fast growing function exceeding Graham's number [duplicate]

If we define $G_0 = 10^{100}$, and $G_n = 10^{G_{n-1}}$ (hence $G_0$ is a googol, $G_1$ is a googolplex, $G_2$ is a googolplexian), for what first value of n will $G_n$ exceed Graham's number?
12
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1answer
431 views

What is $\underbrace{2018^{2018^{2018^{\mathstrut^{.^{.^{.^{2018}}}}}}}}_{p\,\text{times}}\pmod p$ where $p$ is an odd prime?

This recent question inspired me to explore values concerning modulo arithmetic of tetrations, and I thus pose the following question. Is there a general expression for the value of $$\underbrace{...
2
votes
2answers
120 views

Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven?

See here : http://googology.wikia.com/wiki/Arrow_notation for the definition of the up-arrow function. Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven for all $m\ge 1$ and $n\ge 3$ ?...
1
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0answers
55 views

Are those estimates of the magnitude of huge numbers correct?

See here http://googology.wikia.com/wiki/Fast-growing_hierarchy for the definitions of the fast growing hierarchy, chained-arrow-notation and two-dimesnional-array-notation. The first number is ...
2
votes
1answer
305 views

What's the largest proven lower-bound for SCG(13)?

I hope someone can answer this question. If you can answer it, then you already know what SCG(13) is. SCG(13) is a very, very large number which is part of a theorem about graphs. It's at least one ...
0
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0answers
42 views

Can I prove $X\rightarrow 1\rightarrow Y=X$ using the given definition?

Here : http://googology.wikia.com/wiki/Chained_arrow_notation the definition of the chained arrow notation is given. How can I prove $$X\rightarrow 1\rightarrow Y=X$$ for every chains $X,Y$ ...
2
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1answer
147 views

Decimal expansion of the number $f_3(3)$ with PARI/GP?

I tried to calculate the number $f_3(3)$ with PARI/GP. It seems the number is too large to be calculated exactly. I would like to analyze the full decimal expansion. Is there any trick to get all ...
0
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2answers
185 views

3↑↑↑3= ? but with 10 instead of 3 ( approximation, order of magnitude )

3↑↑↑3= (or near) in power tower of 10 or in ( Knuth ) arrow ↑ notation of 10 to get a sense of it's order of magnitude; I grasp numbers more easily with 10 3↑↑↑3 being the first really huge number in ...
4
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2answers
132 views

Magnitude of $f_3(n)$ compared to power towers of tens

In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$ The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$ For example, to calculate $f_3(5)$, we have to ...
10
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1answer
261 views

How did Euler disprove Mersenne's conjecture?

In 1644, Mersenne made the following conjecture: The Mersenne numbers, $M_n=2^n−1$, are prime for $n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257$, and no others. Euler found that the Mersenne ...
7
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2answers
228 views

How to determine $n$, such that $x\uparrow \uparrow n>10^{100}$?

If $x$ is a real number greater than $e^{e^{-1}}$ , then $x\uparrow \uparrow n$ (A power tower of $n$ $x's$) tends to $\infty$, if $n$ tends to $\infty$. Therefore, there must be a number $n$, such ...
1
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1answer
752 views

Why is TREE(3) not infinite? [duplicate]

this is my first ever question on this forum so bear with me if the formatting or phrasing of the question itself seems strange... I was reading about TREE(3) and the rules followed in generating ...
7
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1answer
2k views

Proof that TREE(n) where n >= 3 is finite?

Reading online, it generally seems accepted that TREE(n) where n >= 3 is a finite number, but large enough to be incomputable and only has extremely loose lower bounds today. TREE(n) is the function ...
3
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2answers
718 views

How to solve factorial equations with very big numbers

I have problem. I've calculated memory complexity of my algorithm. In exchange of very good time complexity of my algorithm, I have memory complexity $x!$, where $x$ is number of elements my algorithm ...
4
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0answers
245 views

Sanity check: does $D_{\omega_9}(9)$ exceed TREE(3)?

TREE(3) For the Golf a number bigger than TREE(3) challenge I wrote a program but I'm not sure it is bigger than TREE(3). The function TREE(k) gives the length of the longest sequence of trees T1, ...
1
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1answer
76 views

How does my modified ordinal hierarchy relate to other ordinal hierachies?

I was working on my answer for Golf a number bigger than TREE(3) and I realized I couldn't use The Hardy Hierarchy in the way wanted to. So I defined a slightly modified version: $$H'_0(n)=n+1$$ $$H'_\...
4
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0answers
116 views

Smallest prime factor of $\lfloor e\uparrow e\uparrow e\uparrow e\rfloor$?

What is the smallest prime factor of $$\lfloor e\uparrow e\uparrow e\uparrow e\rfloor$$ To get this number start with $1$ and apply the $\exp$-funtion four times, then take the integer part. This ...
7
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1answer
159 views

Smallest twin-prime-pair above $2\uparrow\uparrow 5\ $?

I searched the smallest prime larger than $$N:=2\uparrow\uparrow 5=2^{65536}$$ $N$ has $19\ 729$ digits. This is quite large and finding primes of this magnitude is not easy any more. I found $$N+44\ ...
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2answers
170 views

Has something like “Knuth's Up-Arrow Factorial Notation” ever been used? If so, what practical uses does it have?

I was studying Knuth's up-arrow notation and I was wondering if ever something like "Knuth's up-arrow factorial notation" has ever been used. Now I know this probably isn't a recognizable term, ...
3
votes
1answer
178 views

The Ackermann hierarchy vs. the fast growing hierarchy

Suppose we've defined the Ackermann hierarchy as follows: $$A_\alpha(n)=\begin{cases}n+1,&\alpha=0\\A_{\alpha[n]}(n),&\alpha\in\Bbb{Lim}\\A_\beta(1),&n=0,\alpha=\beta+1\\A_\beta(A_\alpha(...
2
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1answer
272 views

computable function not outgrowed by fast growing hierarchy

I have been looking at this question Does there exist a computable function that grows faster than fast growing hierarchy?, but I dont understand the answer and cant use it to answer this question ...
4
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1answer
194 views

Tighter bounds on the fast growing hierarchy?

Not a dupe of this question, as I'm searching for tighter bounds. We define the fast growing hierarchy for finite values as follows: $$f_k(n)=\begin{cases}n+1,&k=0\\f_{k-1}^n(n),&k>0\end{...
3
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0answers
88 views

For which n does the nth “hyperoperation number” n[n]n begin with n in base 2?

If $[n]$ denotes the $n$th binary hyperoperation in the sequence $(+,\times,\uparrow,\uparrow\uparrow,...)$, then the following equality is readily verified for $n=1,2,3,4:$ $$n\,[n]\,n\ =\ (n_2...)...
8
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1answer
208 views

Are there nontrivial equations for hyperoperations above exponentiation?

A similar question was asked in comments elsewhere. A paper by Roberto Di Cosmo and Thomas Dufour ("The Equational Theory of 〈ℕ, 0, 1, + , ×, ↑〉 Is Decidable, but Not Finitely Axiomatisable") asserts ...
1
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1answer
60 views

how to get the integer part of very large integer multiples of irrational numbers such as $\pi$?

Do I need exact value of $\pi $ upto say $100$ digits if my multiplier is order $10^{10}$.
0
votes
3answers
473 views

How Big would “Graham's Tree” be?

What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?
2
votes
2answers
109 views

How does one make a large number using computable methods?

Context: I'm starting another contest and I was interested in how one makes an extremely large finite number as simply as possible. For starters, the Ackermann function is extremely simple: $$A(0,n)=...
3
votes
1answer
92 views

What are these asymptotics called when two functions are bounded by a fixed shift of the other?

When studying large numbers and extremely fast growing functions, I've noticed that the normal big-O notations are not enough to reasonably compare things. Instead, I've been using this: $$f=\...
0
votes
3answers
94 views

Are there more permutations of pixels in a picture or bases in the human genome?

An iPhone 7 takes pictures that have roughly 12 Megapixels. For simplicity, let's assert that the picture only encodes 256 values per red, green and blue channels such that a 1x1 pixel image has 256^3 ...
12
votes
1answer
782 views

The first few values of Rayo's function?

Rayo's function defined in English: "$\operatorname{Rayo}(n)$ is the smallest positive integer bigger than any finite positive integer named by an expression in the language of first order set theory ...