Mathematics Stack Exchange
Mathematics Stack Exchange

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.

For questions relating to the handling of large finite numbers. This is related to googology, which is the study and nomenclature of large numbers.

To place a scale, numbers around the size of a googol ($10^{100}$) and larger are considered "large".

275 questions
305
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21 answers

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is…
Justin L.
  • 15,120
105
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2 answers

Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on

Popular mathematics folklore provides some simple tools enabling us compactly to describe some truly enormous numbers. For example, the number $10^{100}$ is commonly known as a googol, and a googol plex is $10^{10^{100}}$. For any number $x$, we…
JDH
  • 45,373
92
votes
5 answers

Is it possible to represent every huge number in abbreviated form?

Consider the following expression. $1631310734315390891207403279946696528907777175176794464896666909137684785971138$ $2649033004075188224$ This is a $98$ decimal digit number. This can be represented as $424^{37}$ which has just 5 digits. or…
endrendum
  • 233
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votes
1 answer

Why is TREE(3) so big? (Explanation for beginners)

I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions. David…
Josh Kerr
  • 957
63
votes
18 answers

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas…
Alec
  • 4,174
48
votes
3 answers

What is the fastest-growing function that's useful in some area of math?

Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly speaking, "increase too quickly" (I'm aware of the…
Michael Blakeman
  • 1,034
37
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4 answers

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) or a 9 followed by nine factorial symbols. Both…
Nathan
  • 373
29
votes
7 answers

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number of solutions? Are there ones with so large…
Vladimir Reshetnikov
  • 32,650
29
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1 answer

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to process in reasonable time with numerical libraries…
Vladimir Reshetnikov
  • 32,650
28
votes
1 answer

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 defined by Harvey Friedman, based on Joseph…
Ivan G.
  • 403
22
votes
1 answer

Graham's Number : Why so big?

Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !
thetruth
  • 1,902
20
votes
1 answer

Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz

The question is motivated by a question on a can of number noodles. Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then you'll get the biggest number if you start with the…
Nikolaj-K
  • 12,559
18
votes
13 answers

What is the fastest growing total computable function you can describe in a few lines?

What is the fastest growing total computable function you can describe in a few lines? Well, not necessarily the fastest - I just would like to know how far an ingenious mathematician can go using only a few lines, and what systematic approaches…
Nik Z.
  • 1
18
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1 answer

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 with $n$ symbols or less." More formally, we make…
Simply Beautiful Art
  • 76,603
16
votes
2 answers

Could someone tell me how large this number is?

Context: If you guys didn't know, I'm running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to join). If you are entering, do note that I am…
Simply Beautiful Art
  • 76,603
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