🧵 The relationship between factorials and tetration

Anonymous at Mon, 15 Jul 2024 23:27:55 UTC No. 16282989

In case any of you are unfamiliar with tetration, it is the next hyperoperation after exponents.
For example 2 tetrated to 5, 25 = 2^2^2^2^2 = 2*10^19728
There is an interesting relationship between tetration and factorials.
Take, x!=yz
(better written as Γ(x+1)=yz)
If y≥2 AND z≥2, then (yz-2) ≤ x ≤ (yz-1) is always true.
Examples:
22=4
x!=4,
x=2
(yz-2): 20=1
(yz-1): 21=2
1 ≤ 2 ≤ 2

24=65536
x!=65536,
x=8.23
(yz-2):22=4
(yz-1):23=16
4 ≤ 8.23 ≤ 16

Take it a bit more extreme:
44 = 10^10^153.9
x!=10^10^153.9,
x=5.2*10^151
(yz-2): 42=256
(yz-1): 43=1.34*10^154
256 ≤ 5.2*10^151 ≤ 1.34*10^154

This goes to show that even for numbers incomprehensibly large, such as 100100, we know that the factorial such that
x!=100100
is somewhere in the magnitude between 10098 and 10099