FANDOM

10,231 Pages

The Clarkkkkson, denoted ¥, is a dynamic googologism that grows over time, based on the lynz.[1][2]

Let $$n!$$ represent the factorial, and create the following extension:

• $$n!! = n! \cdot (n - 1)! \cdot (n - 2)! \cdot (n - 3)! \cdot \ldots$$
• $$n!!! = n!! \cdot (n - 1)!! \cdot (n - 2)!! \cdot (n - 3)!! \cdot \ldots$$
• $$n!!!! = n!!! \cdot (n - 1)!!! \cdot (n - 2)!!! \cdot (n - 3)!!! \cdot \ldots$$
• etc.

For the sake of this article, $$n!!$$ will be abbreviated as $$n!2$$, $$n!!!$$ as $$n!3$$, and so forth. The notation is due to Aalbert Torsius.

Define the hypf(c,p,n) function as follows (using down-arrow notation):

• $$\text{hypf}(c, 2, n) = n!c \cdot (n - 1)!c \cdot (n - 2)!c \cdot \ldots$$
• $$\text{hypf}(c, 3, n) = n!c \downarrow (n - 1)!c \downarrow (n - 2)!c \downarrow \ldots$$
• $$\text{hypf}(c, 4, n) = n!c \downarrow\downarrow (n - 1)!c \downarrow\downarrow (n - 2)!c \downarrow\downarrow \ldots$$
• and so on through the weak hyper-operators.

Further, define the Clarkkkkson function ck(c, p, n, r) = hypf(c, p, ck(c,p,n,r-1)) and ck(c,p,n,1)=hypf(c,p,n).

Finally, define the following series:

• $$A_{1} = ck(K, K, K, K)$$ (where $$K$$ is the lynz)
• $$A_{2} = ck(A_{1}, A_{1}, A_{1}, A_{1})$$
• $$A_{3} = ck(A_{2}, A_{2}, A_{2}, A_{2})$$
• etc.
• The Clarkkkkson is AK + 1.

The Clarkkkkson changes over time. Thus, it can be considered a function of time instead of a true number.

The Clarkkkkson passed Graham's Number shortly after it was defined.

Its current value is approximately fω+1(10102174) (refresh), using the fast-growing hierarchy.