User blog:B1mb0w/The J Function

The J Function
The J function is a reasonably fast growing function. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here|WORK IN PROGRESS for more information about the Alpha Function.

Introduction
For an introduction, it will be useful to refer to my other blogs on the Strong D function. on:

\(d(a,b,c)=d(a-1,d(a,b-1),d(a,b,c-1))\)

\(D(a,b,c)=D(a-1,D(a,b,c-1),D(a,b,c-1))\)

Definition
\(D = 0\) This is a null function that always returns zero.

\(D(3) = 4\) This is the successor function

\(D(1,2) = 5\) This is the same as d(1,2)

\(D(2,3) = 17\) This is the same as d(2,3)

Using the Comparison Rule C1 \(d(m,n) >> f_{m-1}(n+2)\) we get

\(D(2,3) >> f_1(5) = 10\)

\(D(3,4) = 5099 >> f_2(6) = 6.2^6 = 384\)

\(D(3, 9) = 1,240,025 >> 1,000,000\)

\(D(3, 206) = 122*10^{98} >>\) Googol

\(D(4,1) >> f_{3}(3) = f_{\omega}(3)\)

\(D(4,2) >>\) Googolplex

\(D(6,1) >> g_1\) where \(g_64 = G\) Graham's number

Other calculations give these results

\(D(D(4,1)+1,D(4,1)-1) >> f_{\omega}^2(3)\)

\(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(4,4) >> f_3(6) >> f_{\omega}(3)\)

\(D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >>\) very low bound \(>> f_{\omega}^2(3)\)

\(D(1,0,2) >> f_{\omega}^3(3) >> f_{\omega+1}(3)\)

Using Rule: N1 \(D(1,0,n) >> f_{\omega}^{n-2}(f_{\omega+1}(3))\) when n>2

\(D(1,0,3) >> f_{\omega}(f_{\omega+1}(3))\)

\(D(1,0,4) >> f_{\omega}^2(f_{\omega+1}(3))\)

and

\(D(1,0,D(1,0,2)+2) = D(1,0,f_{\omega+1}(3)+2) >> f_{\omega}^{f_{\omega+1}(3)+2-2}(f_{\omega+1}(3))\)

\(= f_{\omega}^{f_{\omega+1}(3)+2-2}(f_{\omega+1}(3)) = f_{\omega}^{f_{\omega+1}(3)}(f_{\omega+1}(3))\)

\(= f_{\omega+1}(f_{\omega+1}(3)) = f_{\omega+1}^2(3)\)

Using Rule: NL \(D(l,m,n) = D(l,0,n-1+(m+2).(m+1)/2)\) or

\(D(1,m,n) = D(1,0,n-1+(m+2).(m+1)/2)\)

\(D(1,9,9) = D(1,6,36) = D(1,3,54) = D(1,0,63) >> g_{64} = G\) Graham's number

and

\(D(1,0,D(1,0,3)) >> D(1,0,D(1,0,2)+2) >> f_{\omega+1}^2(3)\)

Calculated Examples up to D(2,0,n)
Let \(e = D(1,0,D(1,0,3))\) then

\(D(1,0,e+2) >> f_{\omega}^{f_{\omega+1}^2(3)+2-2}(f_{\omega+1}(3)) = f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega+1}(3))\)

\(D(1,0\(D(1,0,f+2) >> f_{\omega}^{f_{\omega+2}(3)}(f_{\omega+1}(3))\)

\(D(1,0,f+e+2) >> f_{\omega}^{f_{\omega+(1,0,D(1,0,D(1,0,4)))+D(1,0,D(1,0,3))+D(1,0,2)+2 = f+e+D(1,0,2)+2\)

\(D(1,0,D(1,0,D(1,0,D(1,0,5)))) >> f_{\omega+1}(f_{\omega+2}(3))\)

and

\(D(1,0,D(1,0,D(1,0,D(1,0,D(1,0,6))))) >> f_{\omega+1}^2(f_{\omega+2}(3))\) also refer to this blog

then

\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >> D(1,0,D(1,0,3)) >> f_{\omega+1}^2(3)\)

\(D(2,0,1) = D(1,D(2,0,0),D(2,0,0)) >> D(1,0,D(2,0,0))\)

\(>> D(1,0,D(1,D(1,2,2),D(1,2,2))) >> D(1,0,D(1,0,D(1,2,2))) >> f_{\omega+1}^2(3)\)

\(>> D(1,0,D(1,0,D(1,0,4))) >> f_{\omega+2}(3)\)

\(D(2,0,2) >> D(1,0,D(2,0,1)) >> D(1,0,D(1,0,D(2,0,0)))\)

\(>> D(1,0,D(1,0,D(1,0,D(1,2,2)))) >> D(1,0,D(1,0,D(1,0,D(1,0,5)))) >> f_{\omega+1}(f_{\omega+2}(3))\)

and

\(D(2,0,3) >> f_{\omega+1}^2(f_{\omega+2}(3))\) also refer to this blog

and

\(D(2,0,n) >> f_{\omega+1}^{n-1}(f_{\omega+2}(3))\) also refer to this blog

Calculated Examples up to D(3,0,n)
\(D(2,0,D(2,0,1)+1) >> f_{\omega+1}^{f_{\omega+2}(3)+1-1}(f_{\omega+2}(3)) = f_{\omega+2}(f_{\omega+2}(3))\)

\(= f_{\omega+2}^2(3)\)

\(D(3,0,0) = D(2,D(2,3,3),D(2,3,3)) >> D(2,0,D(2,0,1)+1) >> f_{\omega+2}^2(3)\)

Let \(g = D(2,0,D(2,0,1))\) then

\(D(2,0,g+D(2,0,1)+1) >> f_{\omega+1}^{g+D(2,0,1)}(f_{\omega+2}(3))\)

\(>> f_{\omega+1}^{f_{\omega+2}^2(3)+f_{\omega+2}(3)}(f_{\omega+2}(3))\)

\(= f_{\omega+1}^{f_{\omega+2}^2(3)}(f_{\omega+2}^2(3)) = f_{\omega+2}^3(3) = f_{\omega+3}(3) = f_{\omega.2}(3)\)

\(D(3,0,1) = D(2,D(3,0,0),D(3,0,0)) >> D(2,0,D(2,D(2,3,3),D(2,3,3))) >> D(2,0,D(2,0,D(2,3,3)))\)

\(D(2,0,D(2,3,3)) >> D(2,0,D(2,0,1))+D(2,0,1)+1 >> g+D(2,0,1)+1\)

then

\(D(3,0,1) >> D(2,0,g+D(2,0,1)+1) >> f_{\omega.2}(3)\)

\(D(3,0,2) = D(2,D(3,0,1),D(3,0,1)) >> D(2,D(3,0,1),0)\)

\(>> D(2,0,D(3,0,1)+D(3,0,0)+D(2,0,1)+1) >> f_{\omega+1}^{f_{\omega.2}(3)+f_{\omega+2}^2(3)+f_{\omega+2}(3)}(f_{\omega+2}(3))\)

\(= f_{\omega+1}^{f_{\omega.2}(3)+f_{\omega+2}^2(3)}(f_{\omega+2}^2(3))\)

\(= f_{\omega+1}^{f_{\omega.2}(3)}(f_{\omega.2}(3)) = f_{\omega+2}(f_{\omega.2}(3))\)

\(D(3,0,n) >> f_{\omega+2}(f_{\omega.2}(3))\)

\(D(1,0,0,0) = D(0,D(0,1,1,1),D(0,1,1,1),D(0,1,1,1)) = D(D(1,1,1),D(1,1,1),D(1,1,1))\)

Next - the Alpha Function
My next blog post will introduce a new Alpha function that I have been thinking about.