Prime Number Spiral
Hermetic Systems

Software for exploring the Ulam Spiral of Prime Numbers

An interesting prime number spiral was discovered in 1963 by Stanislaw M. Ulam, and is now called "the Ulam spiral". It reveals a strange property of the prime numbers.

A positive integer (1, 2, 3, ... ) greater than 1 is called prime if its only divisors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ... The series is infinite, since there is no largest prime number. The proof of this goes back to the ancient Greek mathematician Euclid.

The Ulam spiral of prime numbers is constructed as follows: Consider a rectangular grid. We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise) as at below left. The prime numbers are then marked (here with blue boxes). Since the primes occur in an irregular manner in the sequence of numbers one might expect that in this grid they would occur more or less at random, and so form something like a random pattern. But, on the contrary, there is a clear tendency for the prime numbers to form diagonal lines. This can be seen more clearly in the image below right, in which each of the 70,255 pixels corresponds to a position in the number sequence and the primes are marked by white pixels. This pattern is puzzling, since no complete explanation has been given for why the prime numbers should line up in this way.

Central part of Ulam spiral of prime numbers   Ulam spiral of prime numbers

This Prime Number Spiral software is a tool for exploring the Ulam spiral. Here is a typical screen:

Screenshot showing prime number spiral

The remainder of the user manual is here:
  1. Customization of the display
  1. Ascertaining the number at a location
  2. Square numbers
  3. Triangular numbers
  4. Spiralling in and out
  5. Restricting the central number to be prime
  6. Measuring the longest diagonals
  7. Random numbers
  8. Saving images

How to obtain the software: A fully-functional version of the Prime Number Spiral software is available for free download from this website. Click on the following link to go to a web page with further information:

Download Prime Number Spiral ...