Ray Primes

by Simon Whitechapel

Prime spirals, or Ulam spirals, were first investigated by the Polish-American mathematician Stanisław Ulam (1909-84). In this counter-clockwise spiral beginning at 1, the prime numbers are printed in red:

Here prime numbers are represented as white squares, composite numbers as black squares:

A prime spiral can also be represented as a pattern of 1’s, for primes, and 0’s, for composites (the central 0 represents the number 1, as in the spiral at the top):

Each 1 or 0 can be regarded as the center of eight rays of binary numbers, thus:

Reading clockwise towards the center, starting with the vertical ray, the binary numbers are (the central number is regarded as the first righthand digit of each number):

At no point does the running total of a ray equal 1, the number symbolized by the central 0. However, elsewhere in the prime spiral, the running total of one or more rays does equal the number symbolized by the central 1 or 0. Here is the second ray for 19 = 10011₂:

And here are the fourth and eighth rays for 23 = 10111₂:

19 is consequently a ray prime and 23 a double ray prime. Some composite numbers, such as 256 and 512, are also ray numbers, but the largest ray number I have discovered so far is another prime, 7841 = 1111010100001₂: