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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3
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0answers
39 views

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 ...
3
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2answers
59 views

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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1answer
59 views

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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1answer
36 views

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 ...
3
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1answer
65 views

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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1answer
56 views

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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27 views

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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2answers
49 views

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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1answer
74 views

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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2answers
28 views

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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1answer
64 views

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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2answers
55 views

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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0answers
103 views

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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2answers
100 views

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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1answer
40 views

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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1answer
51 views

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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1answer
69 views

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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1answer
115 views

psychology: why are we more stunned by very big finite numbers than infinite cardinals? [closed]

Reading about Tree(3) and then thinking about Tree(Tree(3)) etc stun me and some other people more than reading about inaccessible cardinals and even larger cardinals. Why might this be?
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1answer
63 views

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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1answer
149 views

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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1answer
94 views

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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83 views

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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1answer
54 views

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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1answer
54 views

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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45 views

Very weak lower bound for TREE 3. [duplicate]

First of all i am not looking for very accurate lower bound. It would be enough to me the following : Prove that $TREE(3)>f_{\omega +2}(3)$ (here that function $f$ comes from fast growing ...
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1answer
124 views

Comparing big numbers.

Let $G_{64}$ is a Graham Number: https://googology.wikia.org/wiki/Graham%27s_number $TREE(3)$ is a particular value of a special sequence $TREE(k)$ https://googology.wikia.org/wiki/TREE_sequence $...
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1answer
67 views

Nested Tetration properties

Regarding tetration, I know properties like ${}^a({}^bn)= {}^{ab}n$ do not hold in general. When $a=b=2$, for instance, we have $$ {}^2({}^2n)={}^2\left(n^n \right)=\left(n^n \right)^{\left(n^n \right)...
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2answers
116 views

Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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126 views

Prove that sums of powers are equal

All numbers between $1$ and $10^{20}$ ($10^{20}$ not included) are divided into two sets: one with numbers, whose sum of digits is odd and the other with remaining numbers(those, whose sum of digits ...
3
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1answer
138 views

Compare power towers

Prove or disprove: $3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours. Taking logs is useless, and there seems to be no other way to compare. Thanks!
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2answers
86 views

Graham's number closest power of two

Using Knuth's up-arrow notation, Graham's number $G$ is defined as $$ G=g_{64},\,\,\, \text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3 \text{ and } g_n=3\uparrow^{g_{n-1}}3. $$ I was wondering if ...
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0answers
34 views

Is there any algorithm for enumeration of all trees in TREE(3) sequence? [duplicate]

What is the reasoning leading to conclusion that the TREE(3) is so big? Is there any algorithm generating all the trees in sequence? I know such algorithm would not finish in meaningfull time, but it ...
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1answer
43 views

Exponent-factorial inequality marathon [closed]

A) I am wondering which one is bigger? $(((5!)!)!)!$ or $5^{5^{5^5}}$. B) And if there is a largest number with at most four $5$’s and four operations, or there is no such number. Here, we define ...
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97 views

What are the last digits of Moser's number?

How can I find the terminating digits of Moser's number, which is 2 in a "mega"-gon in Steinhaus-Moser notation (a mega being 2 in a pentagon)? I was able to calculate the last 6 digits by analyzing ...
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1answer
69 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. ...
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2answers
149 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 $[1,10]$. ...
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2answers
67 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 ...
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1answer
72 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://...
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1answer
239 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 ...
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4answers
144 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 ...
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1answer
432 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 ...
5
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1answer
337 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....
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4answers
1k 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 ...
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1answer
206 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^{\...
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0answers
251 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 ...
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0answers
1k 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?
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4answers
2k 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 ...
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0answers
81 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 $(...
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2answers
263 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
179 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 ...

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