Conway Chained Arrow Notation - Definition and Overview

Definition and Overview

A Conway chain (or chain for short) is defined as follows:

• Any positive integer is a chain of length 1.
• 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 four rules below. Two chains are said to be equivalent if they represent the same integer.

If and are positive integers, and is a subchain, then:

1. The chain represents the number .
2. represents the exponential expression .
3. is equivalent to .
4. is equivalent to
   (with p copies of X, p − 1 copies of q, and p − 1 pairs of parentheses; applies for q > 0).

Note that the last rule can be restated recursively to avoid the ellipses:

4a.

4b.