3.0.2 - Conway's Chained Arrow Notation (1996)
Background:
In 1996 John Horton Conway and Richard K. Guy published a book on recreational mathematics Titled "The Book of Numbers". In this book Conway discusses many things, among them very large numbers, and some notations to express them. He introduces the general reader to Donald Knuth's Up-Arrow Notation, and Graham's Number, then invents his own notation to surpass them. Conway's Chained Arrow Notation is introduced, along with the cg-function defined on them. Conway's Chained Arrow Notation allows the reader to easily surpass Graham's Number, with Conway Chains of at least 4 entries. Conway Chains are the most well known example of a Chain Notation.
A Conway Chain is an Expression of the form:
a-->b-->...-->x -->y-->z
Where a,b, ... ,x,y,z are positive integer values.
A Conway Chain may have any positive integer number of entries.
(1) If a Conway Chain contains exactly 1 entry then:
a = a
(2) If a Conway Chain contains exactly 2 entries then:
a-->b = a^b (a to the power of b)
(3) If a Conway Chain contains at least 3 entries then:
(3a) Check if the last entry is 1 and remove it from the chain:
a-->b-->...-->x-->y-->1 = a-->b-->...-->x-->y
(3b) Check if the second to last entry is 1 and remove the last two entries from the chain:
a-->b-->...-->x-->1-->z = a-->b-->...-->x
(3c) If none of the above applies then:
a-->b-->...-->x-->y-->z
= a-->b-->...-->x-->(a-->b-->...-->x-->(y-1)-->z)-->(z-1)
When "n" is a positive integer, the value cg(n) is defined as:
cg(n) = n-->n-->...-->n-->n
where there are n copies of n