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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(10102047), using the fast-growing hierarchy.

Sources Edit

  1. Further Lynz Explanation and the Clarkkkkson
  2. Notable Properties of Specific Numbers (page 22)

See also Edit

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