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The TRex function generates very large numbers. It has a growth rate approaching \(f_{LVO}(n)\).
The TRex Function is a family of functions \(T()\), \(r()\) and \(e()\) and \(x()\) which use this simple rule set:
\(T(n) = T(0,n) = n + 1\)
\(T(a + 1, n) = T^n(a,n_*)\)
\(T(x(0), n) = T(n,n)\) and other instances of \(n\) can be substituted with \(x(0)\)
\(x(a + 1) = T^{x(a)}(x(a)_*,x(a))\)
and
\(x(1, 0) = x^{x(0)}(0)\)
\(x(1, a + 1) = x^{x(1, a)}(x(1, a))\)
\(x(b + 1, 0) = x^{x(b, 0)}(b, 0_*)\)
\(x(1, 0, 0) = x^{x(1, 0)}(1_*, 0)\)
and
\(e(0) = x(1, 0_{[x(0)]})\)
\(e(a + 1) = T^{e(a)}(e(a)_*,e(a))\)
\(e(1, 0) = e^{e(0)}(0)\)
\(e(1, a + 1) = e^{e(1, a)}(e(1, a))\)
\(e(b + 1, 0) = e^{e(b, 0)}(b, 0_*)\)
\(e(1, 0, 0) = e^{e(1, 0)}(1_*, 0)\)
and
\(r(0) = e(1, 0_{[…
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THIS BLOG HAS BEEN REPLACED BY MY BLOG ON The TRex Function WHICH IS MUCH STRONGER.
The Rex function generates very large numbers. It has a growth rate approaching \(f_{LVO}(n)\).
The Rex Function is a family of functions \(R()\), \(e()\) and \(x()\) which use this simple rule set:
\(R(n) = R(0,n) = n + 1\)
\(R(a + 1, n) = R^n(a,n_*)\)
\(R(x(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(x(0)\)
\(x(a + 1) = R^{x(a)}(x(a)_*,x(a))\)
and
\(x(1, 0) = x^{x(0)}(0)\)
\(x(1, a + 1) = x^{x(1, a)}(x(1, a))\)
\(x(b + 1, 0) = x^{x(b, 0)}(b, 0_*)\)
\(x(1, 0, 0) = x^{x(1, 0)}(1_*, 0)\)
and
\(e(0) = x(1, 0_{[x(0)]})\)
\(e(a + 1) = R^{e(a)}(e(a)_*,e(a))\)
\(e(1, 0) = e^{e(0)}(0)\)
\(e(1, a + 1) = e^{e(1, a)}(e(1, a))\)
\(e(b + 1, 0) = e^{e(b, 0)…
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THIS BLOG HAS BEEN REPLACED BY MY BLOG ON The Rex Function WHICH IS MUCH STRONGER.
The R function generates very large numbers. It is based on my earlier work on the The S Function.
It has a growth rate \(\approx f_{LVO}(n)\).
The R Function is actually two functions \(R()\) and \(r()\) which use this simple ruleset:
\(R(n) = R(0,n) = n + 1\)
\(R(a + 1, n) = R^n(a,n_*)\)
\(R(r(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(r(0)\)
\(r(a + 1) = R^{r(a)}(r(a)_*,r(a))\)
and
\(r(1, 0) = r^{r(0)}(0)\)
\(r(1, a + 1) = r^{r(1, a)}(r(1, a))\)
\(r(b + 1, 0) = r^{r(b, 0)}(b, 0_*)\)
\(r(1, 0, 0) = r^{r(1, 0)}(1_*, 0)\)
and
\(R(1, 0, n) = R(r(1, 0_{[r(0)]}),n)\)
Some R Function identities are:
\(R(R(R(a,b)),b) > R(R(a,b),R(a,b))\)
because
\(R(R…
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The Generalised S function generates very large numbers. It has a growth rate \(\approx f_{LVO}(n)\).
It is the fourth version of my substitution functions. It is based on the earlier three versions but it has the fastest growth rate of them all. Refer to my other blogs for more information on my work.
The Generalised S Function is actually two functions \(S()\) and \(g()\) where the \(S()\) function behaves the same as it does in each of the other versions, but the \(g()\) function has been defined to create a faster growth rate.
The growth rate of the Generalised S Function is beyond \(f_{LVO}(n)\).
The S function is defined recursively as follows:
\(S(a,b,c)\) where \(a,b,c\) can be finite integer \(n\), an \(S()\) function or a \(g()\) func…
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This version of The Alpha Function has been rewritten to use Javascript in Google Sheets. The code is available for anybody to use or copy as they like. This code replaces my last version which used VBA in Microsoft Excel.
The function code is still based on The S Function (Version 2), with a growth rate of \(f_{\varphi(1,1,0)}(n)\).
Version 9 has been completely rewritten to use Javascript in Google Sheets. A link to the first draft Google Sheet file is available here:
First Draft Google Sheet File
Version 9 has also been 'recalibrated' to allow an input parameter range from 0 to 100,000 that should be more interesting. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Javascr…
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