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The Supernova Array is a method of creating very large numbers using recursion of Up-Arrow Notation in a simple linear notation. It is loosely based on the Ultra-Factorial Funcional Array.


Definitions and Examples

  • Base Function: \(S(a,b) = a\uparrow^{(b)}a\), where \(b\) is the number of \(\uparrow\)'s.
  • \(S(a,b,0) = S(a,b)\)
  • \(S(a,b,c) = S(a,S(a,b),c-1)\)
  • \(S(a,b,c,0) = S(a,b,c)\)
  • \(S(a,b,c,d) = S(a,b,S(a,b,c),d-1)\)
  • \(S(a,b,c,...x,y,z) = S(a,b,c,...x,S(a,b,c,...x,y),z-1)\)

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 \(S(a,b)\).


\(S(1,1) = 1\uparrow 1 = 1\)

\(S(2,2) = 2\uparrow \uparrow 2 = 4\)

\(S(3,3) = 3\uparrow \uparrow \uparrow 3\) = tritri

\(S(10,10) = 10\uparrow^{(10)} 10\) = tridecal

\(S(3,1,1) = S(3,S(3,1),0) = S(3,S(3,1)) = S(3,3\uparrow 3) = S(3,27)\)

\(S(3,3,1) = S(3,S(3,3)) = S(3,tritri)\)

\(S(10,10,1) = S(10,S(10,10)) = S(10,tridecal)\)

\(S(3,3,3) = S(3,S(3,3),2) = S(3,tritri,2) = S(3,S(3,tritri),1) = S(3,S(3,S(3,tritri)),0)\)

\(S(3,4,63) = S(3,S(3,4),62) = S(3,S(3,S(3,4)),61)...\) = Graham's Number


Extended Notations

In extended notations, parentheses are added into the array in order to apply systems of recursion.

  • \(S((a,b)) = S(a,a,a,...a)\), where there are \(b\) number of \(a\)'s
  • \(S((a,b,c,...x,y,z)) = S((a,b,c,...x,S(a,b,c,...x,y),z-1))\)
  • \(S(((a,b))) = S((a,a,a...,a))\), where there are \(b\) number of \(a\)'s
  • \(S(((a,b,c,...x,y,z))) = S(((a,b,c,...x,S(a,b,c,...x,y),z-1)))\)


In general, \(S(a,b)_n = S(a,a,a,...a)_{n-1}\), where there are \(b\) number of \(a\)'s and \(n\) number of paretheses. Alternatively, \(S(((a,b)))\) can be written as \(S(a,b)_3\).

There may be cases wherein a simplified notation may be desirable, as in \(S(a,b)_{S(a,b)}\). In this case, a subscript is added to S, such that the aforementioned function can be rewritten as \(S_1(a,b)\). In general,

  • \(S_0(a,b) = S(a,b)\)
  • \(S_{(n+1)}(a,b) = S(a,b)_{(S_n(a,b))}\), where there are \(S_n(a,b)\) number of parentheses in \(S(((...(a,b)))...)\)
  • \(S_n(a,b,c,...x,y,z) = S_n(a,b,c,...x,S(a,b,c,...x,y),z-1)\)


Likewise, a simplified notation may be desired for functions such as \(S_{S(a,b)}(a,b)\). A superscript is instead assigned to S, such that the aforementioned function may be rewritten as \(S^1(a,b)\). In general,

  • \(S^0(a,b) = S(a,b)\)
  • \(S^{(n+1)}(a,b) = S_{(S^n(a,b))}(a,b)\)
  • \(S^n(a,b,c,...x,y,z) = S^n(a,b,c,...x,S(a,b,c,...x,y),z-1)\)


Furthermore, \(S^{S(a,b)}(a,b)\) could also use a more simplified notation. A subscript is then added BEFORE S, such that the aforementioned function may be rewritten as \(_1S(a,b)\). In general,

  • \(_0S(a,b) = S(a,b)\)
  • \(_{(n+1)}S(a,b) = S^{(_nS(a,b))}(a,b)\)
  • \(_nS(a,b,c,...x,y,z) = _nS(a,b,c,...x,S(a,b,c,...x,y),z-1)\)


Finally, a function such as \(_{(S(a,b))}S(a,b)\) would also benefit from simplification. A superscript is then added BEFORE S, such that the aforementioned function may be rewritten as \(^1S(a,b)\). In general,

  • \(^0S(a,b) = S(a,b)\)
  • \(^{(n+1)}S(a,b) = _{(^nS(a,b))}S(a,b)\)
  • \(^nS(a,b,c,...x,y,z) =  ^nS(a,b,c,...x,S(a,b,c,...x,y),z-1)\)


However, the function's limit has been reached with the pre-superscript. In order to further expand the function, a new more succint formulation has to be added - \(S(a,b)[n,k]\).

  • \(S(a,b)_n\) can be rewritten as \(S(a,b)[n,0]\),
  • \(S_n(a,b)\) can be rewritten as \(S(a,b)[n,1]\),
  • \(S^n(a,b)\) can be rewritten as \(S(a,b)[n,2]\), and so on.


In general,

  • If \(k = 0\), \(S(a,b)_n\ = S(a,b)[n,0] = S(a,a,a,...a)[n-1,0]\), where there are \(b\) number of \(a\)'s
  • If \(k > 0\) and \(n = 0\), \(S(a,b)[0,k] = S(a,b)\)
  • If \(k > 0\) and \(n > 0\), \(S(a,b)[n,k] = S(a,b)[S(a,b)[n-1,k],k-1]\)


Although this function can already produce extremely huge numbers, it can still be further extended as follows:

\(S(a,b)[n,k,0] = S(a,b)[n,k]\)

\(S(a,b)[n,k,p] = S(a,b)[n,S(a,b)[n,k],p-1]\)

\(S(a,b)[n,k,p,...x,y,z] = S(a,b)[n,k,p,...x,S(a,b)[n,k,p,...x,y],z-1]\)

\(S(a,b)[ [n,k] ] = S(a,b)[n,n,n,...n]\), where there are \(k\) number of \(n\)'s

\(S(a,b)[n,k][1,0] = S(a,b)[n,k][0,s] = S(a,b)[n,k]\)

\(S(a,b)[n,k][r,0] = S(a,b)[n,n,n,...n][r-1,0]\), where there are \(k\) number of \(n\)'s

\(S(a,b)[n,k][r,s] = S(a,b)[n,k][S(a,b)[n,k][r-1,s],s-1]\)

\(S(a,b)[n,k][r,s,0] = S(a,b)[n,k][r,s]\)

\(S(a,b)[n,k][r,s,t] = S(a,b)[n,k][r,S(a,b)[n,k][r,s],t-1]\)

\(S(a,b)[n,k][r,s,t,...x,y,z] = S(a,b)[n,k][r,s,t,...x,S(a,b)[n,k][r,s,t,...x,y],z-1]\)

\(S[0](a,b) = S(a,b)\)

\(S[\gamma](a,b) = S[\gamma -1](a,b)[a,a][a,a][a,a]...[a,a]\), where there are \(b\) number of \(a\)'s

\(S[\gamma](a,b)[n,k] = S[\gamma](a,b)[S(a,b)[n-1,k],k-1]\)

Etc.

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