# 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.

171 questions
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### $2014^{2014^{2014}}\bmod2011$ [closed]

I get a question on exam today. What is $$2014^{2014^{2014}}\bmod2011$$ I know that the answer is $985$ but how do I get it?
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### 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 $n\ge 3$ ?...
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### 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 ...
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### 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 ...
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### 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 $X,Y$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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, ...
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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: $$f=\... 3answers 89 views ### Are there more permutations of pixels in a picture or bases in the human genome? An iPhone 7 takes pictures that have roughly 12 Megapixels. For simplicity, let's assert that the picture only encodes 256 values per red, green and blue channels such that a 1x1 pixel image has 256^3 ... 1answer 742 views ### The first few values of Rayo's function? Rayo's function defined in English: "\operatorname{Rayo}(n) is the smallest positive integer bigger than any finite positive integer named by an expression in the language of first order set theory ... 1answer 106 views ### Factorial and exponential relationships (Problem) I have faced some problems like: x,A\in N 32!=A*10^x \Rightarrow Max(x)=?\\ 26!=A*3^x \Rightarrow Max(x)=? My question will be stated after solving the first one as following: Since 10=2*5 ... 2answers 109 views ### How are the first digits of these numbers calculated?! According to googology.wikia, we have the following:$$5^{4^{3^{2^{1}}}}=620606987866087447074832055728467\ldots6^{5^{4^{3^{2^{1}}}}}=110356022591769663217914533447534\ldots How are the ...
Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the ...
Does there exist a computable function that grows faster than fast growing hierarchy for every computable ordinal $\alpha$? Or does it follow that fast growing hierarchy grows as fast as any ...