FANDOM


The Factorial Recursion System (FRS) is a system of notation designed to produce very large numbers using factorials as its base.

Definitions and Examples

  • \(n\iota 0 = n!\)
  • \(n\iota (a+1) = n!^{n\iota a}\), where there are \(n\iota a\) number of !s


  • \(2\iota 0 = 2! = 2\)
  • \(3\iota 0 = 3! = 6\)
  • \(2\iota 1 = 2!^{2\iota 0} = 2!^{2!} = 2!! = 2\)
  • \(3\iota 1 = 3!^{3\iota 0} = 3!^{3!} = 3!!!!!!\)
  • \(3\iota 2 = 3!^{3\iota 1} = 3!^{3!^{3!}} = 3!^{3!!!!!!}\)
  • \(3\iota 3 = 3!^{3\iota 2} = 3!^{3!^{3!^{3!}}} = 3!^{3!^{3!!!!!!}}\) = ultra-3\(\alpha\)

Extended Notation

The extended notation is applied similarly to Up-arrow 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 \(a\) number of \(n\)'s
  • If \(a = 0, n\iota \iota \iota... \iota 0 = n \iota 0 = n!\)


  • \(3\iota \iota 0 = 3! = 6\)
  • \(3\iota \iota 1= 3 \iota 1 = 3!!!!!!\)
  • \(3\iota \iota 2 = 3\iota 3\) = ultra-3\(\alpha\)
  • \(3\iota \iota 3 = 3\iota 3\iota 3 = 3\iota (3\iota 3) = 3\iota\)ultra-3\(\alpha\)...
  • \(3\iota \iota \iota 3 = 3\iota \iota 3\iota \iota 3 = 3\iota \iota (3\iota \iota 3)\) = ultra-3\(\beta\)


More Extensions

  • \(n\iota a = n\iota_0 a\)
  • \(n\iota_{c+1} a = n\iota_c \iota_c \iota_c... \iota_c n\), where there are \(n\iota_c a\) number of \(\iota_c\)’s


  • \(3\iota_1 3 = 3\iota \iota \iota... \iota 3\), where there are \(3\iota 3\) number of \(\iota\)'s
  • \(3\iota_1 \iota_1 3 = 3\iota_1 3\iota_1 3\)
  • \(3\iota_3 3 = 3\iota_2 \iota_2 \iota_2... \iota_2 3\), where there are \(3\iota_2 3\) number of \(\iota_2\)'s

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