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Graham's Number is a huge number, and it is has been difficult to write its exact value despite the numerous notations already developed for very large numbers. Famous functions such as the fastgrowing hierarchy and chained arrow notation only come up with approximations, while the Graham Array Notation and G function give the exact value using complex definitions. The Supernova Array offers an alternative method of writing Graham's Number with a simple set of rules.
 Base Function: \(S(a,b) = a\uparrow^{(b)}a\), where \(b\) is the number of \(\uparrow\)'s.
 \(S(a,b,0) = S(a,b)\)
 \(S(a,b,c) = S(a,S(a,b),c1)\)
The other rules are available here.
We all know that \(3\uparrow \uparrow \uparrow \uparrow 3 = g_1\), \(g_2 = 3 \uparrow^{(g_1)} 3\) andâ€¦
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The Supernova Array is a method of creating very large numbers using recursion of UpArrow Notation in a simple linear notation. It is loosely based on the UltraFactorial Funcional Array.
 Base Function: \(S(a,b) = a\uparrow^{(b)}a\), where \(b\) is the number of \(\uparrow\)'s.
 \(S(a,b,0) = S(a,b)\)
 \(S(a,b,c) = S(a,S(a,b),c1)\)
 \(S(a,b,c,0) = S(a,b,c)\)
 \(S(a,b,c,d) = S(a,b,S(a,b,c),d1)\)
 \(S(a,b,c,...x,y,z) = S(a,b,c,...x,S(a,b,c,...x,y),z1)\)
In general, replace the second to the last input (\(y\)) with the entire function excluding the last input (\(z\)), then subtract 1 from the last input. Do the same for the function within, and the subsequent function, and the function after that, etc. until you reach the base function \(S(a,b)\).
\(S(1â€¦
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The UltraFactorial Functional Array (UFA) is a method of creating very large numbers using factorials and recursion in a simple linear notation.
 \(a,b = a!^{(b)}\), where \(b\) is the number of \(!\)'s.
 \(1,0 = 1\)
 \(1,1 = 1! = 1\)
 \(2,1 = 2! = 2\)
 \(3,1 = 3! = 6\)
 \(2,2 = 2!! = 2\)
 \(3,2 = 3!! = 6! = 720\)
 \(4,2 = 4!! = 24! = 620,448,401,733,239,439,360,000\)
 \(3,3 = 3!!! = 6!! = 720!\)
In extended notation, another input is added into the array in order to operate a system of recursion.
 \(a,b,0 = a,b\)
 \(a,b,c = a,a,b,c1\)
In general, if \(c\) is a positive integer, replace \(b\) with \(a,b\) and subtract 1 from \(c\). Repeat as necessary until you reach the base function \(a,b\).
 \(3,1,0 = 3,1 = 3! = 6\)
 \(3,1,1 =â€¦
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The Factorial Recursion System (FRS) is a system of notation designed to produce very large numbers using factorials as its base.
 \(n\iota 0 = n!\)
 \(n\iota (a+1) = n!^{n\iota a}\), where there are \(n\iota a\) number of !s
 \(2\iota 0 = 2! = 2\)
 \(3\iota 0 = 3! = 6\)
 \(2\iota 1 = 2!^{2\iota 0} = 2!^{2!} = 2!! = 2\)
 \(3\iota 1 = 3!^{3\iota 0} = 3!^{3!} = 3!!!!!!\)
 \(3\iota 2 = 3!^{3\iota 1} = 3!^{3!^{3!}} = 3!^{3!!!!!!}\)
 \(3\iota 3 = 3!^{3\iota 2} = 3!^{3!^{3!^{3!}}} = 3!^{3!^{3!!!!!!}}\) = ultra3\(\alpha\)
The extended notation is applied similarly to Uparrow notation:
 \(n\iota \iota a = n\iota (n\iota (n\iota... n\), where there are \(a\) number of \(n\)'s
 \(n\iota \iota \iota a = n\iota \iota (n\iota \iota\ (n\iota \iota ... n\), where there are \â€¦
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A very fastgrowing hierarchy (VFGH) is a modification of the fastgrowing hierarchy (FGH) designed to compute for even larger numbers than its predecessor.
The following definitions of the VFGH are identical to the FGH:
 \(v_0(n) = n + 1\)
 \(v_\alpha(n) = v_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal
However, in order to achieve an even faster growth rate, the other definition has been modified with an additional level of iteration: \(v_{\alpha+1}(n) = v^{v_{\alpha}(n)} _\alpha(n)\), where \(v^{v_{\alpha}(n)}\) denotes function iteration
In general: \(v_1(n) = 2n + 1\)
 \(v_2(n) = (2^{2n+1})(n+1)  1\)
 \(v_3(n) = unknown\)
Following the aforementioned rules and generalities:
 \(v_0(1) = 1 + 1 = 2\)
 \(v_0(2) = 2 + 1 = 3\)
 \(v_1(1) = 2*1 â€¦
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