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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 \(n\geq 0\), 

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

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