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*10^19728There 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=4x!=4,x=2(yz-2): 20=1(yz-1): 21=21 ≤ 2 ≤ 224=65536x!=65536,x=8.23(yz-2):22=4(yz-1):23=164 ≤ 8.23 ≤ 16Take it a bit more extreme:44 = 10^10^153.9x!=10^10^153.9, x=5.2*10^151(yz-2): 42=256(yz-1): 43=1.34*10^154256 ≤ 5.2*10^151 ≤ 1.34*10^154This goes to show that even for numbers incomprehensibly large, such as 100100, we know that the factorial such that x!=100100is somewhere in the magnitude between 10098 and 10099
>>16282989I had up arrows in the original post text to indicate tetration but they got removed when I posted it :(
>>16282992Here's a legible version
>>16283014If x! = 100 ^^ 100, then 100 ^^ 98 < x! < 100 ^^ 99 is a contradiction.
>>16282989what is a "gamma" function and what's it for?
>>16282989Gamma function is a generalization of the factorial but not factorial itself, faggot. Factorial is a discrete function
