Conway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. . As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power. Definition and overview A "Conway chain" is defined as follows: Any positive integer is a chain of length . A chain of length n, followed by a right-arrow → and a positive integer, together form a chain of length . Any chain represents an integer, according to the six rules below. Two chains are said to be equivalent if they represent the same integer. Let denote positive integers and let denote the unchanged remainder of the chain. Then: An empty chain (or a chain of length 0) is equal to The chain represents the number . The chain represents the number . The chain represents the number (see Knuth's up-arrow notation) The chain represents the same number as the chain Else, the chain represents the same number as the chain . Properties A chain evaluates to a perfect power of its first number Therefore, is equal to is equivalent to is equal to is equivalent to (not to be confused with ) Interpretation One must be careful to treat an arrow chain as a whole. Arrow chains do not describe the iterated application of a binary operator. Whereas chains of other infixed symbols (e.g. 3 + 4 + 5 + 6 + 7) can often be considered in fragments (e.g. (3 + 4) + 5 + (6 + 7)) without a change of meaning (see associativity), or at least can be evaluated step by step in a prescribed order, e.g. 34567 from right to left, that is not so with Conway's arrow chains. For example: The sixth definition rule is the core: A chain of 4 or more elements ending with 2 or higher becomes a chain of the same length with a (usually vastly) increased penultimate element. But its ultimate element is decremented, eventually permitting the fifth rule to shorten the chain. After, to paraphrase Knuth, "much detail", the chain is reduced to three elements and the fourth rule terminates the recursion. Examples Examples get quite complicated quickly. Here are some small examples: (By rule 2) (By rule 3) Thus, (By rule 4) (By rule 4) (see Knuth's up arrow notation) (By rule 4) (see tetration) (By rule 6) (By rule 3) (By rule 5) (By rule 6) (By rule 6) (By rule 4) = much larger than previous number (By rule 6) (By rule 3) (By rule 5) (By rule 6) (By rule 4) = much, much larger than previous number Systematic examples The simplest cases with four terms (containing no integers less than 2) are: (equivalent to the last-mentioned property) We can see a pattern here. If, for any chain , we let then (see