Talk:Clarkkkkson

Originally defined Clarkkkkson is actually between \( 10 \uparrow\uparrow\uparrow 2 \) and \( 10 \uparrow\uparrow\uparrow 3 \), as weak operators can't provide even pentational growth. Ikosarakt1 (talk) 21:01, January 7, 2013 (UTC)


 * It's sort of a salad number FB100Z &bull; talk &bull; contribs 22:38, January 7, 2013 (UTC)

Wait, no. I reconsider that and conclude that it grows pentationally however, and actual value is between tritri and g1. I am interested for more tight bounds, despide that it's just "salad" number. Ikosarakt1 (talk) 13:47, January 14, 2013 (UTC)

Some weak upper bounds and stuff:
 * \(n! < n^n\)
 * \(n!! < n^n \cdot (n - 1)^{n - 1} \cdot (n - 2)^{n - 2} \ldots = (n^n)^n = n^{n^2}\)
 * \(n!!! < \left(n^{n^2}\right)^n = n^{n^3}\)
 * \(n!c < n^{n^c}\)
 * \(\text{hypf}(c, 2, n) < n^{n^c} \cdot (n - 1)^{(n - 1)^c} \ldots < n^{n \cdot n^c} = n^{n^c} \uparrow n\)
 * \(\text{hypf}(c, 3, n) < n^{n^c} \uparrow (n - 1)^{(n - 1)^c} \uparrow \ldots < n^{n^c} \uparrow\uparrow n\)
 * \(\text{hypf}(c, p, n) < n^{n^c} \uparrow^{p - 2} (n - 1)^{(n - 1)^c} \uparrow^{p - 2} \ldots < n^{n^c} \uparrow^{p - 1} n < n^{n^c} \uparrow^{p - 1} n^{n^c}\)
 * \(\text{ck}(c, p, n, r) < n^{n^c} \uparrow^p 2^r\)
 * \(f(a) = a \uparrow^c a \implies f^n(a) \leq a \uparrow^{c + 1} 2^n\). I'm too lazy to prove this, but it "looks" right.
 * \(\text{ck}(n, n, n, n) < n^{n^n} \uparrow^n 2^n\)
 * \(n^{n^n} \uparrow^n 2^n < n \uparrow^{n + 1} n\) (rather sketchy comparison)
 * \(\text{ck}^{n}(n) < n \uparrow^{n + 2} 2^n\)
 * ¥ < {K,, K + 2}

The variables have to be sufficiently large, but on order of the lynz the bounds will definitely work. I'm convinced that ¥ << Graham's number.

Also, I assumed the strong hyper-operators instead of the weak ones, so these bounds are not very tight. Good enough for government work, as they say. FB100Z &bull; talk &bull; contribs 00:14, January 15, 2013 (UTC)