efficient way to express large numbers

Question

I recently watched the walkthrough of Graham's Number on YouTube (Numberphile). Mind-blowing of course.

I then puttered around in other large number topics like Ackerman and Tree(3) and fast growing hierarchies. It's all fantastic.

I also came across a challenge that was to write out or describe the largest number you can in a succinct way. So I thought of a rule and I was wondering if any of you large number smarties could take a look at it and see how it compares to other things.

I'll call the number $V(n)$. It's not the solution to anything, its only virtue is its simplicity.

$V(n)$ is generated in $n$ steps. 1) Start with a power tower of $n$ that is $n$ high. Evaluate. 2)use previous result as the base and height of the next tower. Repeat until $n$ steps.

$V(1)$: $1^1 = 1$

$V(2)$:

• $2^2 = 4$
• $4^{4^{4^4}} = 4^{1.34 \times 10^{154}}$ (Hey, it's bigger than Tree(2)!)

$V(3)$:

• $3^{3^3} = 7.6$ trillion
• tower with base and height of 7.6 trillion (= MONSTER)
• tower with base and height of MONSTER

etc.

Anyone with the chops to compare this with other biggies?

Thanks!

Victor

PS could not find a good tag for this question!

Answer 1

If we define $f(n) = n \uparrow\uparrow n$ using Knuth's up-arrow notation, then $f(n)$ is an exponential tower of $n$'s of height $n$ (This is known as "tetration"). Then $V(n) = f^n(n)$, that is, $V(n)$ is $f$ applied to $n$ a total of $n$ times. This is bigger than $n \uparrow\uparrow\uparrow n$, since the base keeps growing, but it is not much bigger; for example, $V(n)$ will be less than $2 \uparrow\uparrow\uparrow 2n$, although I will not try to prove that here. So $V(n)$ is has similar growth rate to triple arrows in Knuth notation, which is also known as "pentation". Pentation appears for example in the best upper bound to the numbers derived from the Hales-Jewett Theorem. Of course, numbers of the form $V(n)$ using reasonable values of $n$ will be much less than Graham's Number or the Moser, since those are numbers where the number of arrows required to upper bound them are themselves large numbers that require Knuth notation, and therefore much more than three! And of course Graham's Number is nothing compared to TREE(3).

Another useful notation is the fast-growing hierarchy, defined by

$F_0 (n) = n+1$

$F_{\alpha+1}(n) = F_{\alpha}^n(n)$

Note that the fast-growing hierarchy can be extended to infinite ordinal subscripts, but for now I will only talk about finite subscripts. One can show that $F_1(n) = 2n, F_2(n) = n 2^n$, and $F_3(n) > n \uparrow \uparrow n$. So we will have $F_4(n) = F_3^n(n) > f^n(n) = V(n)$. So $V(n)$ is less than (but comparable to) the fourth level of the fast-growing hierarchy.

Answer 2

Well, to try to make your function easier to write, we could use Knuth's arrow-notation, as mentioned by Deedlit.

$$V(n)=V(n-1)\uparrow\uparrow V(n-1)=V(n-1)\uparrow\uparrow\uparrow 2$$$$V_1(n)=n\uparrow\uparrow n$$

Why we would write it like this? Because it is simply easier to understand.

And since you haven't defined your number as, say, $V(10)$, I will have to make an assumption that your question is on how does your number-creation-algorithm compare to the way other big numbers are made.

Simply put, it is much, much, much smaller than Graham's number. Why?

Think about my function, $f(a,b,c)=a\uparrow^bc$.

Let us compare the following is sizes:

$f(5,5,6),f(5,6,5),f(6,5,5)$

We quickly see the following:

$$f(6,5,5)<f(5,5,6)<f(5,6,5)$$

What does this tell you?

For my function, increasing $b$ has the largest effect. This effect is much larger than increasing $a$ or $c$.

So the way we get to Graham's number is as follows:

$$G=f(3,G_{63},3)$$$$G_n=3\uparrow^{G_{n-1}}3, G_0=4$$

We quickly see that we are increasing the $b$ term of my function, and by a LOT.

So your numbers are small compared to Graham's number, which is small compared to many other "big numbers".

Why did I choose to compare it to Graham's number? It uses the same notation, the same pattern(ish), and is therefore relatively easy to compare.

So far, it is rather difficult to compare most "big numbers".