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The Ultra-Factorial Functional Array (UFA) is a method of creating very large numbers using factorials and recursion in a simple linear notation.


Definitions and Examples

  • \(|a,b| = a!^{(b)}\), where \(b\) is the number of \(!\)'s.


  • \(|1,0| = 1\)
  • \(|1,1| = 1! = 1\)
  • \(|2,1| = 2! = 2\)
  • \(|3,1| = 3! = 6\)
  • \(|2,2| = 2!! = 2\)
  • \(|3,2| = 3!! = 6! = 720\)
  • \(|4,2| = 4!! = 24! = 620,448,401,733,239,439,360,000\)
  • \(|3,3| = 3!!! = 6!! = 720!\)


Extended Notation

In extended notation, another input is added into the array in order to operate a system of recursion.

  • \(|a,b,0| = |a,b|\)
  • \(|a,b,c| = |a,|a,b|,c-1|\)

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 \(|a,b|\).


  • \(|3,1,0| = |3,1| = 3! = 6\)
  • \(|3,1,1| = |3,|3,1|,0| = |3,6| = 3!!!!!!\)
  • \(|3,2,1| = |3,|3,2|,0| = |3,720| = 3!!!...!\), where there are 720 number of \(!\)'s
  • \(|3,3,1| = |3,|3,3|,0| = |3,720!| = 3!!!...!\), where there are 720! number of \(!\)'s
  • \(|3,3,2| = |3,|3,3|,1| = |3,720!,1| = |3,|3,720!|,0| = |3,|3,720!|| = 3!!!...!\), where there are |3,720!| number of \(!\)'s


Further Extensions

In further extensions, more inputs are simply added into the array to put in place even more recursion.

  • \(|a,b,c,0| = |a,b,c|\)
  • \(|a,b,c,d| = |a,b,|a,b,c|,d-1|\)
  • \(|a,b,c,d,0| = |a,b,c,d|\)
  • \(|a,b,c,d,e| = |a,b,c,|a,b,c,d|,e-1|\)
  • \(|a,b,c,...x,y,z| = |a,b,c,...x,|a,b,c,...x,y|,z-1|\)

In general, replace the second to the last input (in this case, \(y\)) with the entire function excluding the last input (in this case, \(z\)), 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 \(|a,b|\).


  • \(|3,3,3,0| = |3,3,3| = |3,|3,3|,2| = |3,720!,2| = |3,|3,720!|,1|\)...
  • \(|3,1,1,1| = |3,1,|3,1,1|,0| = |3,1,|3,|3,1||| = |3,1,|3,6||\)
= \(|3,1,3!!!!!!|\)
= \(|3,|3,1|,3!!!!!!-1|\)
= \(|3,6,3!!!!!!-1|\)
= \(|3,|3,6|,3!!!!!!-2|\)
= \(|3,|3,3!!!!!!|,3!!!!!!-2|\)
= \(|3,3!!!...!,3!!!!!!-2|\)...

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