clarkkkkson

The "clarkkkkson" is an exotic number which was defined by a bunch of school kids. It also happens to be bigger than Graham's number.

The definition and construction of large numbers is an interesting branch of mathematics. For one thing, it hardly qualifies as mathematics. No number larger than Graham's number has yet found itself a mathematical use. Therefore, there is no real use for numbers ranging between Graham's number and the lowest infinite ordinals and cardinals. As such, defining and naming numbers in this range is more of a feat of creativity and imagination than it is a mathematical necessity. This makes it an amusingly pointless pursuit, but at the same time, accessible to less mathematically inclined individuals, if only in terms of "Wow, that's a big number"-type sentiments.

The clarkkkkson is named after one Andrew Clarkson, and is most thoroughly defined on this page. The story itself not worth repeating, but it is an interesting exercise to show that it exceeds Graham's number. Here, I will provide a quick definition followed by my proof.

Note that Graham's number can be constructed briefly as:

G1 = hyper(3,6,3)
Gn+1 = hyper(3,Gn+2,3)
G64 = Graham's Number

where hyper() is the hyper operator.

Brief construction of the clarkkkkson (henceforth C)

Define the "multifactorial" function (denoted by multiple exclamation marks) recursively as:

       k=a
a!   =  Π  k, as with the regular factorial
       k=1

           k=a
a!!!...! =  Π  k!!!...!
 \_____/   k=1  \_____/
    b             b-1

For example:

1! = 1
6! = 6 * 5 * 4 * 3 * 2 * 1
6!! = 6! * 5! * 4! * 3! * 2! * 1!
6!!! = 6!! * 5!! * 4!! * 3!! * 2!! * 1!!
etc.

Now define the "hyperfactorial" function hypf as

hypf(a,b,c) = c!!!...! ↑↑↑...↑ (c-1)!!!...! ↑↑↑...↑ ... ↑↑↑...↑ 2!!!...! ↑↑↑...↑ 1!!!...!
               \_____/ \_____/      \_____/ \_____/     \_____/  \_____/ \_____/  \_____/
                  a      b-2           a      b-2         b-2       a      b-2       a

where the up-arrows work as in Donald Knuth's arrow notation. For example:

hypf(1,1,1) = 1!
hypf(1,1,6) = 6! ↑ 5! ↑ 4! ↑ 3! ↑ 2! ↑ 1!
hypf(1,2,6) = 6! ↑↑ 5! ↑↑ 4! ↑↑ 3! ↑↑ 2! ↑↑ 1!
hypf(2,2,6) = 6!! ↑↑ 5!! ↑↑ 4!! ↑↑ 3!! ↑↑ 2!! ↑↑ 1!!
etc.

Define the "clarkkkkson function" ck as

ck(a,b,c,1) = hypf(a,b,c)
ck(a,b,c,d) = hypf(a,b,ck(a,b,c,d-1))

For example:

ck(2,2,2,2) = hypf(2,2,ck(2,2,2,1))
            = hypf(2,2,hypf(2,2,2))
            = hypf(2,2,2!! ↑↑ 1!!)
            = hypf(2,2,2)
            = 2
ck(3,3,3,3) = hypf(3,3,ck(3,3,3,2))
            = hypf(3,3,hypf(3,3,ck(3,3,3,1)))
            = hypf(3,3,hypf(3,3,hypf(3,3,3)))
            = hypf(3,3,hypf(3,3,3!!! ↑↑↑ 2!!! ↑↑↑ 1!!!))
            = hypf(3,3,hypf(3,3,24 ↑↑↑ 2))
            = hypf(3,3,hypf(3,3,24 ↑↑ 24))
            = hypf(3,3,hypf(3,3,24 ↑ 24 ↑ ... ↑ 24))
            = ...

Then finally define the "alpha function" (my own creation to simplify the working) as

A(a) = ck(a,a,a,a)

Then the clarkkkkson is equal to

C = AK(K)

where K is the number of "lynz" (lines) which Andrew Clarkson is expected to write out. K is actually a variable, K(t), which has increased rapidly in real time since the lynz were first assigned in the late 1990s. K was equal to one googolplex (10¹⁰¹⁰⁰) at 6:34:15am on August 9th, 1999 and squares every 24 hours. At the time of writing it has squared 2748 times since then, making it equal to roughly 10¹⁰⁹²⁷.

Proof that the clarkkkkson exceeds Graham's number

First note that A, ck, hypf and hyper are all strictly increasing in all their arguments.

Establishing a lower bound on the hypf function

Proof that the clarkkkkson exceeds Graham's number first involves establishing a relation between the somewhat nonstandard "hyperfactorial" function and the more conventional hyper operator, as follows:

hypf(a,b,c) = c!!!...! ↑↑↑...↑ (c-1)!!!...! ↑↑↑...↑ ... ↑↑↑...↑ 2!!!...! ↑↑↑...↑ 1!!!...!
               \_____/ \_____/      \_____/ \_____/     \_____/  \_____/ \_____/  \_____/
                  a      b-2           a      b-2         b-2       a      b-2       a

            > c ↑↑↑...↑ (c-1) ↑↑↑...↑ ... ↑↑↑...↑ 2 ↑↑↑...↑ 1
                \_____/       \_____/     \_____/   \_____/
                  b-2           b-2         b-2       b-2

            > c ↑↑↑...↑ (c-1)
                \_____/
                  b-2

            = hyper(c,b,c-1)

Construction

Now consider A(6).

A(6) = ck(6,6,6,6)
     = hypf(6,6,ck(6,6,6,5))
     > hyper(ck(6,6,6,5),6,ck(6,6,6,5)-1)
     > hyper(3,6,3)
     = G1

Now inductively, suppose A^k(6) > G_n. Then

Ak+2(6) = A(A(Ak(6)))
       > A(A(Gn))
       = A(ck(Gn,Gn,Gn,Gn))
       = A(hypf(Gn,Gn,ck(Gn,Gn,Gn,Gn-1)))
       > A(hyper(ck(Gn,Gn,Gn,Gn-1),Gn,ck(Gn,Gn,Gn,Gn-1)-1))
       > A(hyper(3,Gn,3))
       = ck(hyper(3,Gn,3),hyper(3,Gn,3),hyper(3,Gn,3),hyper(3,Gn,3))
       > ck(3,hyper(3,Gn,3),3,3)
       = hypf(3,hyper(3,Gn,3),ck(3,hyper(3,Gn,3),3,2))
       > hypf(3,hyper(3,Gn,3),4)
       > hyper(4,hyper(3,Gn,3),3)
       > hyper(3,Gn+2,3)
       = Gn+1

Thus A^2n-1(6) > G_n for all n, and specifically A¹²⁷(6) > G₆₄ = Graham's number.

Finally,

AK(K) > A127(K) > A127(6) > G64

QED.

So what?

Well, here's where it all comes from. xkcd is a web comic. It is a very geeky, mathematically-inclined webcomic. Here is a typical strip. Note the number defined in the third panel - the Ackermann function with Graham's number as arguments.

Having stumbled across the clarkkkkson here, the xkcd author asked in his blag which is bigger, the "clarkkkkson" or what he calls the "xkcd number".

My calculations have led me to conclude that the clarkkkkson is the larger of the two. It's just another step from the above proof but I'll leave it to you to figure out.

So, yeah. It's a pretty silly way to define a number. It's very much rooted in high school mathematics, has no actual application other than to wow the uninformed, and nobody is ever going to put it in a dictionary of mathematics anytime soon. Still. It gave me a few minutes of amusement, which I felt I would share.