Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers.
[1] 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.
A "Conway chain" is defined as follows: Any chain represents an integer, according to the six rules below.
Two chains are said to be equivalent if they represent the same integer.
denote positive integers and let
denote the unchanged remainder of the chain.
denote sub-chains of length 1 or greater.
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 get quite complicated quickly.
The simplest cases with four terms (containing no integers less than 2) are: We can see a pattern here.
The Ackermann function can be expressed using Conway chained arrow notation: hence Graham's number cannot be expressed in Conway chained arrow notation, but it is bounded by the following:
Proof: We first define the intermediate function
, which can be used to define Graham's number as
(The superscript 64 denotes a functional power.)
Since f is strictly increasing, which is the given inequality.
With chained arrows, it is very easy to specify a number much greater than Graham's number, for example,
Conway and Guy created a simple, single-argument function that diagonalizes over the entire notation, defined as:
This function, as one might expect, grows extraordinarily fast.
Peter Hurford, a web developer and statistician, has defined an extension to this notation:
All normal rules are unchanged otherwise.
is much faster growing than Conway and Guy's
are different numbers; a chain must have only one type of right-arrow.
become legal, but the notation as a whole becomes much stronger.