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The hyper-leviathan number is defined like so:[1]

  1. Let \(|1|_1(x) = x\).
  2. Let \(|1|_n(x) = \prod^{x}_{i = 1} |1|_{n - 1}(i) = x!n\)
  3. Let \(|2|_1(x) = |1|_x(x) = T(x)\) using the Torian.
  4. Let \(|2|_n(x) = \prod^{x}_{i = 1} |2|_{n - 1}(i)\)
  5. Let \(|3|_1(x) = |2|_x(x)\).
  6. Let \(|3|_n(x) = \prod^{x}_{i = 1} |3|_{n - 1}(i)\)
  7. Continue in this fashion. Now define \(||1||_1(x) = |x|_x(x)\).
  8. Let \(||1||_n(x) = \prod^{x}_{i = 1} ||1||_{n - 1}(i)\)
  9. Let \(||2||_1(x) = ||1||_x(x)\).
  10. Let \(||2||_n(x) = \prod^{x}_{i = 1} ||2||_{n - 1}(i)\)
  11. Let \(||3||_1(x) = ||2||_x(x)\).
  12. Let \(||3||_n(x) = \prod^{x}_{i = 1} ||3||_{n - 1}(i)\)
  13. Continue in this fashion. Now define \(|||1|||_1(x) = ||x||_x(x)\).
  14. Let \(|||1|||_n(x) = \prod^{x}_{i = 1} |||1|||_{n - 1}(i)\)
  15. Let \(|||2|||_1(x) = |||1|||_x(x)\).
  16. Let \(|||2|||_n(x) = \prod^{x}_{i = 1} |||2|||_{n - 1}(i)\)
  17. Let \(|||3|||_1(x) = |||2|||_x(x)\).
  18. Let \(|||3|||_n(x) = \prod^{x}_{i = 1} |||3|||_{n - 1}(i)\)
  19. Continuing in this fashion, the hyper-leviathan number is \(\underbrace{|||\ldots|||}_{10^{666}}10^{666}\underbrace{|||\ldots|||}_{10^{666}}{}_{10^{666}}\left(10^{666}\right)\).

Size analysisEdit

\(|1|_2(x) = |1|_1(x)*|1|_1(x-1)...|1|_1(2)*|1|_1(1) = x!2 < x^{x^2}\).

\(|1|_n(x) = |1|_{n-1}(x)*|1|_{n-1}(x-1)...|1|_{n-1}(2)*|1|_{n-1}(1) = x!n < x^{x^n}\).

\(|2|_1(x) = x!x = T(x) < x^{x^x}\).

\(|2|_2(x) < x!(x+1) < x^{x^{x+1}}\).

\(|3|_1(x) < x!(x2) < x^{x^{x2}}\).

\(|4|_1(x) < x!(x3) < x^{x^{x2}}\).

\(||1||_1(x) < x!(x^2) < x^{x^{x^2}} \).

\(||2||_1(x) < x!(x^2+x) < x^{x^{x^2+x}}\).

\(|||1|||_1(x) < x!(x^22) < x^{x^{x^22}}\).

\(||||1||||_1(x) < x!(x^23) < x^{x^{x^23}}\).

\(||...||1||...||_1(x) < x!(x^3) < x^{x^{x^3}}\).

So the Hyper-leviathan number is smaller than \(10^{10^{10^{2001}}}\)

Sources Edit

  1. [1]

See also Edit

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