# 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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### 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 ...
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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1 vote
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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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### 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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1 vote
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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 ...
1 vote
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}+...
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