## FANDOM

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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 $$10^{10^{10^{2,001}}}$$.

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