Wilson Primes

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One last post related to the number 13. 

English mathematician John Wilson (1741-1793) developed a theorem for identifying prime numbers (numbers that can be divided only by one and itself). You can play along and try this yourself: 

  1. Start with any natural (whole, positive) number. (Let's say 7.)
  2. Multiply together every number less than your starting number. (If the number you picked was 7, then multiply together 6*5*4*3*2*1, which equals 720.)
  3. Add 1 (totalling 721.)
  4. Divide by the original number (721 divided by 7.)
  5. If the result is a natural number, then the original number that you started with is a prime number. (In our example, we end up with the natural number 103, meaning 7 is a prime number.)

Neat, right? Wilson also identified a special kind of number, where the result was divided not by the original number, but by the square root of the original number. So instead, do the following:

  1. Start with any natural number.
  2. Multiply together every number less than your starting number. 
  3. Add 1.
  4. Divide by the square root of the original number.
  5. If the result is a natural number, then the original number that you started with is a Wilson Prime number. 

Wilson himself could find only two numbers that fit this condition: 5 and 13. For more than 150 years, nobody found any additional Wilson Primes. It did involve, afterall, multiplying together at least hundreds of numbers, and even the mathematicians probably had better things to do. Then, in the 1950s, the idea of using computers to search for significant numbers came along. In 1953, Karl Goldberg used this technique to find a third Wilson Prime: 563. 

Since then, computer processing power has increased rapidly, and in particular, distributed computing methods are perfect for these sort of tasks. The search for Wilson Prime Numbers has now passed the number 400,000,000. No additional Wilson Primes have been found. It's impossible to say when the next Wilson Prime will be found. It could be next week, or it may not happen in our lifetimes. 

In theory, the number of Wilson Primes should be infinite, and the fact that they're so rare doesn't change this theory at all. It's sort of a staggering idea, that something can be so rare, and yet infinite, at the same time. Its inclusion in this series is one of the reasons I've got a new-found admiration for the number 13.