User blog:Wythagoras/SuperJedi's X function

Trivial bounds (X(0) to X(12))
X(0) = 0

X(1) = 9

X(2) = 99

X(3) = 9^9

X(4) = 9^99

X(5) = 9^9^9

X(6) = 9^9^99

X(7) = 9^9^9^9,

X(8) = 9^9^9^99,

X(9) ≥ 9^9^9^9^9,

X(10) ≥ 9^9^9^9^99,

X(11) ≥ 9^9^9^9^9^9,

X(12) ≥ 9^9^9^9^9^99

X(13)
I found a lower bound for X(13) that is slightly better than 9^^7 that could be obtained by the method above. Ax=x^x^x AAA9 X(13) ≥ AAA9 ~ 9^9^9^9^9^(9^9+9).

X(14)
Ax=x^x^x AAAA9 X(14) ≥ AAAA9 ~ 9^9^9^9^9^9^9^(9^9+9).

X(15)
Ax=x^x^x AAAAA9 AAAAA9 ~ 9^9^9^9^9^9^9^9^9^(9^9+9) ~ 9^^11.

X(16)
Ax=x^x^x AAAAAA9 AAAAAA9 ~ 9^9^9^9^9^9^9^9^9^9^9^(9^9+9) ~ 9^^13.

X(17)
A(0)=9 A(Sx)=9^A(x) A(A(9)) A(A(9)) = 9^^(9^^10+1)

X(18)
A(0)=9 A(Sx)=9^A(x) A(A(A(9))) A(A(A(9))) = 9^^(9^^(9^^10+1)+1)

X(19)
A(0)=9 A(Sx)=9^A(x) A(A(A(A(9))) A(A(A(A(9)))) ~ 9^^^5

X(20)
A(0)=9 A(Sx)=9^A(x) A(A(A(A(A(9)))) A(A(A(A(A(9))))) ~ 9^^6

X(21) to X(27)
Continue in the same manner.

X(28)
A(0)=B(0)=9 A(Sx)=9^A(x) B(Sx)=A(B(x)) B(B(9)) B(B(9)) ~ 9^^^9^^^10.

X(29)
A(0)=B(0)=9 A(Sx)=9^A(x) B(Sx)=A(B(x)) B(B(B(9))) B(B(B(9))) ~ 9^^^9^^^9^^^10.

X(30) and X(31)
You know how.

X(32)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(9,9)

X(33)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(99,9)

X(34)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(A(9,9),9)

X(35)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(A(99,9),9)

X(36)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(A(A(9,9),9),9)

X(37)
A(0,y)=Sy A(Sx,0)=A(x,1) A(Sx,Sy)=A(x,A(Sx,y)) A(A(A(99,9),9),9)