Prime Number Spiral
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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.



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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



Compatible with Windows 7 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



Virus-free Demo version: A copy of the Prime Number Spiral program is available for free download from this website for the purpose of evaluation. Click on the following link for further information:
Download Prime Number Spiral ...


Languages: Prime Number Spiral is bilingual: English and German.

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Price and ordering: The price for a single-user license is given at Purchase a User License. An activation key is required in order to make the trial version permanently fully functional. An activation key can be obtained immediately if you purchase a user license either via PayPal or via Share-it.

Refund: A refund will be provided promptly up to 30 days after purchase if the software does not perform satisfactorily.

Updates: Purchasers of a user license for this software are entitled to an update to any later version at no additional cost.

Prime Number Spiral (German)
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