The Prime Number Maze

This paper introduces a fascinating maze based solely on the distribution of the prime numbers. Although it was originally designed as a simple puzzle, the maze revealed some rather startling properties of the primes. The rules are so simple and natural that traversing the maze seems more like exploring a natural cave formation than a maze of human design. We will describe this maze using the language of graph theory. In particular, we first define an undirected graph G0 with the set of all prime numbers as the vertex set. There will be an edge connecting two prime numbers iff their binary representations have a Hamming distance of 1. That is, two primes are connected iff their binary, representations differ by exactly one digit. The natural starting point is the smallest prime, 2 = 102. Following the graph GQ amounts to changing one binary digit at a time to form new prime numbers. The following sequence demonstrates how we can get to larger and larger prime numbers by following the edges of G0.