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Javascript code for my Alpha Function based on my Quantum Function is available in Google Sheets.
You are free to use and copy the code as you like. Version 2.0 of the Javascript code in Google Sheets is available here:
Alpha Function Javascript
I have started working on a new version. Here is some work in progress:
\(\alpha(2.75) = Q(1,3) = 7\)
\(\alpha(3.2) = Q^{2}(1,3_*) = 15\)
\(\alpha(3.61) = Q^{3}(1,3_*) = 31\)
\(\alpha(3.76) = Q^{Q(3)}(1,3_*) = 63\)
\(\alpha(3.95) = Q^{Q^{2}(3)}(1,3_*) = 127\)
\(\alpha(4.14) = Q^{Q^{3}(3)}(1,3_*) = 255\)
\(\alpha(4.3) = Q(2,3) = 511\)
\(\alpha(6.65) = Q^{3}(2,3_*)\)
\(\alpha(8.714) = Q^{Q^{Q^{3}(3)}(1,3_*)}(2,3_*)\)
\(\alpha(8.825) = Q(t_0(0),3)\)
\(\alpha(9.55914) = Q(t_0(1),3)\)
\(\alpha(10.39809) = Q(t_0(2),3)\…
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Javascript code for my Alpha Function based on my Quantum Function is available in Google Sheets.
You are free to use and copy the code as you like. Version 2.0 of the Javascript code in Google Sheets is available here:
Alpha Function Javascript
This version replaces my previous Version 1.0 spreadsheet, but all versions are still available online.
Some examples taken from the spreadsheet.
\(\alpha(2) = Q(1,2) = 5\)\(\alpha(2.75) = Q^{2}(1,2_*) = 11\)
\(\alpha(3.1) = Q^{Q(2)}(1,2_*) = 23\)
\(\alpha(3.61) = Q^{Q^{2}(2)}(1,2_*) = 47\)
\(\alpha(4.29) = Q(t_0(0),2) = 95\)
\(\alpha(5.8721) = Q(Q(1,t_0(0)),2)\)
\(\alpha(8.825) = Q(Q(t_0(0),t_0(0)),2)\)
\(\alpha(11.3623) = Q(Q(Q^{t_0(0)}(t_0(0)),t_0(0)),2)\)
\(\alpha(15.2982) = Q(Q(Q(1,t_0(0)),t_0(0)),2)\)
\(\…
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Javascript code for my Alpha Function based on my Quantum Function is available in Google Sheets.
You are free to use and copy the code as you like. Version 1.0 of the Javascript code in Google Sheets is available here:
Alpha Function Javascript
Some examples taken from the spreadsheet.
\(\alpha(0) = 0\)
\(\alpha(1) = 1\)
\(\alpha(1.375) = 2\)
\(\alpha(1.5) = Q(2) = 3\)
\(\alpha(2) = Q(1,2) = 5\)
\(\alpha(3.01) = Q^{Q^{2}(2)}(Q^{2}(1,2_*)) = Q^{4}(Q^{2}(1,2_*))\)
\(\alpha(4.286) = Q(t_0(0),2)\)
\(\alpha(4.3582) = Q^{Q^{Q^{2}(2)}(1,2_*)}(t_0(0),2_*)\)
\(\alpha(4.3625) = Q(Q(t_0(0)),2)\)
\(\alpha(4.44) = Q(Q^{t_0(0)}(t_0(0)),2)\)
\(\alpha(4.605) = Q(Q(1,t_0(0)),2)\)
\(\alpha(4.9695) = Q(Q(t_0(0),t_0(0)),2)\)
\(\alpha(5.0687) = Q(Q(Q(t_0(0)),t_0(0)),2)\)
\(\a…
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This should be my final attempt to define the Alpha Function. This blog replaces my previous attempt that can be found here
The difference in this new attempt is that the Alpha Function has been redefined using my new Quantum Function which is the fastest growing function I have created.
My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(2)\) for example can be used to reference the number 3. Therefore 2 is the Alpha Index for the number 3.
The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. It is monotonically increasing and every input real \(a > b\), results in a larger output number, where \(\alpha(a) >= \alpha(b)\) in all cases. The fu…
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This blog extends the notation I use that is explained here.
The extension allows the recursion parameter subscript \(*\) to be used within the parameter subscript brackets. This allows concise notation for strongly recursive functions.
This example of Parameter Subscript Brackets is the basis of the first example of Recursion Subscript within the brackets:
\(M(a,0_{[2]}) = M(a,0,0)\)
Normally the Recursion Parameter Subscript in my notation is used to identify the parameter that is nested, as in this example:
\(M^2(a,b_*) = M(a,M(a,b))\)
Applying the Recursion subscript within the brackets significantly increases the scope of recursion by exploding the number of parameters (and therefore the amount of nested recursion required).
\(M^2(a,0_{[*]})…
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