## FANDOM

10,825 Pages

Proofs not included. Some of the following have formal proof, some don't.

1. $$(a\uparrow^nb)\uparrow^nc=a\uparrow^n(b\times c)$$

The equation holds when n=0 or 1 (n=0 for multiplication)

It's a simple property but it doesn't have a name.

This is useful for extention to rational numbers.

$$a\uparrow^n(q/p)=(a\uparrow^nq)\uparrow^n{1/p}$$

Also, $$(a\uparrow^n{1/p})\uparrow^np=a$$, so $$a\uparrow^n{1/p}=a\downarrow_R^np$$

Therefore, $$a\uparrow^n(q/p)=(a\uparrow^nq)\downarrow_R^np$$

(Notation explanation in 10.)

2.$$(a\uparrow\uparrow b)\uparrow\uparrow c<a\uparrow\uparrow (b+c)$$ for $$a\geq2, b\geq1, c\geq1$$

Sadly that it has been proven by Sbiis Saibian (even in more general case)

3.$$lim_{n→≈} a\uparrow\uparrow n$$ exists when $$e^{-e}\leq a\leq e^{1/e}$$ (proven by Euler)

$$lim_{n→≈} a\uparrow^3 n$$ exists when $$1\leq a\leq \text{about} 1.65$$ (by guess)

4. the "a" in $$slog_ab$$ must be bigger than 1 and should not be $$b^{1/b}$$, while the "b" should be in the range of the function $$y=a\uparrow\uparrow x$$

5.for all $$n \geq 2$$, $$y=a\uparrow^n x$$ is strictly increasing and $$a>1$$

6.for all $$n \geq 1$$, $$y=a\uparrow^n x$$ passes (0,1), (-1,0), ..., (1-n,2-n), n points in total. All on the magic line of y=x+1

Define that  $$K_n$$ is the biggest solution smaller than zero to the equation $$a\uparrow^n x=x$$

7.$$lim_{x→-∞} a\uparrow^n x=K_n$$ for odd integer n

8.$$lim_{x→(slog_aK_{n-2})-1} a\uparrow^n x=-∞$$ for even integer n

9.$$-n≤K_n<1-n$$(equals when n=0) and $$-n≤(slog_aK_{n-2})-1<1-n$$(equals when n=2)

10. In my notation :

$$a\uparrow^nb=c$$

$$c\downarrow_L^na=b$$

$$c\downarrow_R^nb=a$$

L for "Left" and "Logarithm"

R for "Right" and "Root"

for all $$n\geq 0$$,

$$b\downarrow_L^na=1+(b\downarrow_L^{n-1}a)\downarrow_L^na$$