FANDOM


1-Goodstein Sequences - G(n,1)

G(n,1) is same at G(n,3), but uses addition.

G(2,1)

X,2

2,3

1,4

0,5

G(3,1)

X+1,2

X,3

3,4

2,5

1,6

0,7

G(4,1)

X+X,2

X,5

0,11

G(5,1)

X+X+1,2

X+X,3

X,7

0,15

So G(n*2+1,1) = 2^{n+2}-1

G(6,1)

X+X+X,2

X+X,5

X,11

0,23

So G(n*2,1) = 3*2^n-1

2-Goodstein Sequences - G(n,2)

G(n,2) is same at G(n,3), but uses addition and multiplication.

G(4,2)

X*X, 2

X*2+2, 3

X*2, 5

X, 11

0, 23

G(5,2)

X*X+1, 2

X*X, 3

X*3+3, 4

X*3, 7

X*2, 15

X, 31

0, 63

G(6,2)

X*(X+1), 2

X*X+2, 3

X*X, 5

X*5+5, 6

X*5, 11

X*4, 23

0, 287

G(7,2)

X*(X+1)+1, 2

X*(X+1), 3

X*X, 7

X*6, 15

0, 1023

G(8,2)

X*(X*X), 2

X*(X*2+2)+2, 3

X*(X*2+2), 5

X*(X*2+1), 11

X*(X*2), 23

X*(X+23)+23, 24

X*(X+23), 47

X*(X+22), 71

X*X, 24*2^24-1

0, (24*2^24)*2^{24*2^24}-1

G(9,2)

X*(X*X)+1, 2

X*(X*X), 3

X*(X*3+3)+3, 4

X*(X*3+3), 7

X*(X*3), 63

X*(X*2+63)+63, 64

X*(X*2), f(63) = 64*2^(64)-1

X*X, f(f(63))

0, f(f(f(63))) = f(f(f(f(3))))

G(10,2)

X*(X*X+1), 2

X*(X*X)+2, 3

X*(X*X), 5

X*(X*4), f(5)

0, f(f(f(f(f(5)))))

3-Goodstein Sequences - G(n)

G(n,3) is same at G(n).

G(4)

X^X, 2

X^2*2+X*2+2, 3

X^2*2+X*2, 5

X^2*2+X, 11

X^2*2, 23 = f(2)

X^2, f(f(3)) = 24*2^24-1 = 3*2^27-1

X*402653183+402653183, 3*2^27

0, 3*2^402653211-1

G(5)

X^X+1, 2

X^X, 3

X^3*3+X^2*3+X*3+3, 4

X^3*3+X^2*3+X*3, 7

X^3*3+X^2*3+X*2, 15

X^3*3+X^2*3+X, 31

X^3*3+X^2*3, 63 = f(3)

X^3*3+X^2*2, 64*2^64-1 = 2^70-1 = f(f(3))

X^3*3+X^2, f^3(3)

X^3*3, f^4(3)

X^3*2+X^2*f^4(3)+X*f^4(3)+f^4(3), f^4(3)+1

X^3*2+X^2*f^4(3)+X*f^4(3), f^4(3)*2-1

X^3*2+X^2*f^4(3), f^5(3)

X^3*2, f^{f^4(3)+4}(3)

X^3, f^{f^4(3)+5}(3)

0, f^{f^{f^4(3)+5}(3)+f^4(3)+6}(3)

G(6)

X^X+X, 2

X^X, 5

X^5*5+X^4*5+X^3*5+X^2*5+X*5+5, 6

X^5*5+X^4*5+X^3*5+X^2*5+X*5, 11

X^5*5+X^4*5+X^3*5+X^2*5, f(5)

X^5*5+X^4*5+X^3*5, f^6(5) = X_1

X^5*5+X^4*5+X^3*4, f^{f^6(5)+1}(f^6(5)) = X_2

X^5*5+X^4*5+X^3*3, f^{f^{f^6(5)+1}(f^6(5))+1}(f^{f^6(5)+1}(f^6(5))) = X_3

X^5*5+X^4*5, X_6 = X_{0,1}

X^5*5+X^4*4+X^3*X_6+X^2*X_6+X*X_6+X_6, X_6+1

X^5*5+X^4*4+X^3*X_6+X^2*X_6, f(X_6)

X^5*5+X^4*4+X^3*X_6, f^{X_6+1}(X_6) = X_{1,1}

X^5*5+X^4*4, f^{X_6+1}(X_6) = X_{X_6+1,1} = X_{0,2}

X^5*5, X_{0,6} = X_{0,0,1}

X^5*5, X_{0,X_{0,6}+1,1} = X_{0,0,2}

0, X_{0,0,6}

G(8)

X^(X+1), 2

X^X*2+X^2*2+X*2+2, 3

X^X*2+X^2*2+X*2, 5

X^X*2+X^2*2, 23

X^X*2, f(f(f(2)))

X^X+X^f(f(f(2)))*f(f(f(2)))+...+X*f(f(f(2)))+f(f(f(2))), f(f(f(2)))+1

X^X+X^f(f(f(2)))*f(f(f(2)))+...+X^2*f(f(f(2))), f(f(f(f(2))))

X^X+X^f(f(f(2)))*f(f(f(2)))+...+X^3*f(f(f(2))), f^{f(f(f(2)))+1}(f(f(f(2))))

4-Goodstein Sequences - G(n,4)

G(n,4) is same at G(n,3), but uses addition, multiplication, exponentation, and tetration.

Introduction to G3(4,4)

X^^3, X^X^X, X^X^3, X^(X^2*X), X^(X^2*3), X^(X^2*2+X*2+3), X^(X^2*2+X*2+2)*3, X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*3, ..., X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X*2+2

So G3(4,4) = X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X*2+2

G(4,4)

X^^X, 2

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X*2+2, 3

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X*2, 5

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X^2*2, 23

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X^2, f(23)

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X^3*2, f(f(23)) = f(f(f(3))) = X

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X^3, f^{X+1}(X) = Y

X^(X^2*2+X*2+2)*2+X^(X^2*2+X*2+1)*2+...+X^X*2, f^{Y+1}(Y)

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.