Mathematical Giants!

[mathbalarka]

How can numbers be large enough? Well, that is easy enough to answer. Take the $\zeta$-function and note that it has a pole at $1$. Oh, no, wrong kind of answer - take the successor function from Peano axiom and note that any bijection $f$ from $\mathbb{Z}$ to $\{0, 1, \cdots, n\}$ must satisfy $f(n + 1) = f(k)$ for some $k \in \{0, 1, \cdots, n\}$, thus violating surjection and contradicting bijection property.

Nevertheless, not every large number is interesting. For example, "quattuordecillion" or $10^{45}$ is not listed in Collin's English dictionary (11th ed.), Cambridge online dictionary or in Oxford English dictionary (2nd ed. and new ed.).

The purpose of this thread is to simply list the large numbers, which are interesting enough. Everyone can post, and the rules are

1. The number must be interesting enough.
2. The number must be large enough (although "enough" can be ruled out if it is very interesting)
3. The number should be computable.
4. Existence of such numbers is not enough to be posted here, and ...
5. The number should be explicitly written here, although abbreviations are accepted, i.e., $10^{100}$ (gogol).
6. If possible, a short intro on the number mentioned would be nice.

My favorites are :

• Gogolplex, $10^{10^{100}}$. Larger than the number of Plank spaces that composes the observable universe.
• Monster number, the order of the Monster group : $808017424794512875886459904961710757005754368000000000$
• Skewes' number $e^{e^{e^{79}}}$, introduced by Skewes as an upper bound to the smallest occurring of $\pi(x) > \text{li}(x)$
• Not-so-large Johnston number $14228$. In Nathaniel's opinion, this is the first uninteresting number since this is the smallest with the property that no OEIS entry contains this.

Feel free to add more,
Balarka
.

 

[Random Variable]

I remember a thread about Graham's number on the old Math Help Forum.

Using up-arrow notation it is defined recursively as $G = g_{64}$ where $g_{1}=3 \uparrow \uparrow \uparrow \uparrow 3 $ and $g_{n} = 3 \uparrow^{g_{n-1}} 3$.


Graham's number - Wikipedia, the free encyclopedia

Knuth's up-arrow notation - Wikipedia, the free encyclopedia

 

[mathbalarka]

Quite nice! Although I much like to use $\text{sexp}^5_3(3)$ rather than Knuth's. (usually $\text{ksexp}$ for Kneser's analytic continuation of tetration)

 

[Deveno]

When I learned about TREE(3), I sort of lost interest in large numbers....

TREE sequence - Googology Wiki

 

[Ackbach]

• Monster number, the order of the Monster group : $808017424794512875886459904961710757005754368000000000$

My abstract algebra and number theory professor said, of this number, "It's cheating to call that finite."

 

[Klaas van Aarsen]

For example, "quattuordecillion" or $10^{45}$ is not listed in Collin's English dictionary (11th ed.), Cambridge online dictionary or in Oxford English dictionary (2nd ed. and new ed.).

Doesn't that make it interesting?
Especially since it's a weird omission, presumably a mistake, as the higher numbers are listed?

 

[Jameson]

This is the biggest number that I have ever heard of that has a purported relevance to math or physics. The Poincaré recurrence theorem states that given enough time certain systems will return to their original state. If you make certain assumptions about our universe and frame the assertion in a certain way, it can be argued that this idea applies to it.

The number is \(\displaystyle 10^{10^{10^{10^{10^{1.1}}}}}\)

Just above this link (scroll up a tiny bit) is a short explanation of the number. My knowledge of physics is poor behind the introductory college level so I can't write more on this number but I find it very interesting.

 

[Deveno]

Some nice visualizations of Poincare recurrence can be seen here:

Arnold's cat map - Wikipedia, the free encyclopedia

This is a discrete example, but differential equations produce similar behavior under certain constraints.

 

[topsquark]

Well, I'm professionally not very interested in anything bigger than a proton so I guess my largest number is about a femtometer. On the other hand protons are expected to have a half-life of \(\displaystyle 10^{34}\) years, so I guess that's a good one too.

-Dan