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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.

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