Hyper-leviathan number

The hyper-leviathan number is defined like so:
# Let \(|1|_1(x) = x\).
# Let \(|1|_n(x) = \prod^{x}_{i = 1} |1|_{n - 1}(i) = x!n\)
# Let \(|2|_1(x) = |1|_x(x) = T(x)\) using the Torian.
# Let \(|2|_n(x) = \prod^{x}_{i = 1} |2|_{n - 1}(i)\)
# Let \(|3|_1(x) = |2|_x(x)\).
# Let \(|3|_n(x) = \prod^{x}_{i = 1} |3|_{n - 1}(i)\)
# Continue in this fashion. Now define \(||1||_1(x) = |x|_x(x)\).
# Let \(||1||_n(x) = \prod^{x}_{i = 1} ||1||_{n - 1}(i)\)
# Let \(||2||_1(x) = ||1||_x(x)\).
# Let \(||2||_n(x) = \prod^{x}_{i = 1} ||2||_{n - 1}(i)\)
# Let \(||3||_1(x) = ||2||_x(x)\).
# Let \(||3||_n(x) = \prod^{x}_{i = 1} ||3||_{n - 1}(i)\)
# Continue in this fashion. Now define \(|||1|||_1(x) = ||x||_x(x)\).
# Let \(|||1|||_n(x) = \prod^{x}_{i = 1} |||1|||_{n - 1}(i)\)
# Let \(|||2|||_1(x) = |||1|||_x(x)\).
# Let \(|||2|||_n(x) = \prod^{x}_{i = 1} |||2|||_{n - 1}(i)\)
# Let \(|||3|||_1(x) = |||2|||_x(x)\).
# Let \(|||3|||_n(x) = \prod^{x}_{i = 1} |||3|||_{n - 1}(i)\)
# Continuing in this fashion, the hyper-leviathan number is \(\underbrace{|||\ldots|||}_{10^{666}}10^{666}\underbrace{|||\ldots|||}_{10^{666}}{}_{10^{666}}\left(10^{666}\right)\).
== Analysis==
\(|1|_2(x) = |1|_1(x)*|1|_1(x-1)...|1|_1(2)*|1|_1(1) = x!2 < x^{x^2}\).
\(|1|_n(x) = |1|_{n-1}(x)*|1|_{n-1}(x-1)...|1|_{n-1}(2)*|1|_{n-1}(1) = x!n < x^{x^n}\).
\(|2|_1(x) = x!x = T(x) < x^{x^x}\).
\(|2|_2(x) < x!(x+1) < x^{x^{x+1}}\).
\(|3|_1(x) < x!(x2) < x^{x^{x2}}\).
\(|4|_1(x) < x!(x3) < x^{x^{x2}}\).
\(||1||_1(x) < x!(x^2) < x^{x^{x^2}} \).
\(||2||_1(x) < x!(x^2+x) < x^{x^{x^2+x}}\).
\(|||1|||_1(x) < x!(x^22) < x^{x^{x^22}}\).
\(||||1||||_1(x) < x!(x^23) < x^{x^{x^23}}\).
\(||...||1||...||_1(x) < x!(x^3) < x^{x^{x^3}}\).
So the hyper-leviathan number is smaller than \(10^{10^{10^{2,001}}}\).
== Sources ==
== See also ==
*Beast number
*Leviathan number
Category:Numbers
Category:Class 4
Category:Even numbers
ja:超リヴァイアサン数