Loading ...
Sorry, an error occurred while loading the content.

Twin primes simple theorem

Expand Messages
  • Kermit Rose
    Here is a simple theorem related to twin primes. I wonder how many times it s been replicated. If the positive integer d cannot be equal to abs( [ (3* m + 1)
    Message 1 of 1 , Dec 18, 2007
    View Source
    • 0 Attachment
      Here is a simple theorem related to twin primes.

      I wonder how many times it's been replicated.

      If the positive integer d cannot be equal to abs( [ (3* m + 1) * n
      + ( 3* n + 1) * m ] )

      for integers m and n, both nonzero,

      then

      both

      6 * d -1 and 6 * d + 1 are prime.



      Note that m and n are permitted to take on both positive and negative
      values, but not zero.


      Illustration:


      m n (3*m+1)*n (3*n+1)*m sum

      1 1 4 4 8
      -1 -1 2 2 4
      1 -1 -4 -2 -6

      It's not possible for 1, 2, 3, or 5 to be of this form,


      so both 6 * 1 - 1 and 6 * 1 + 1 are prime.
      both 6 * 2 -1 and 6 * 2 + 1 are prime
      both 6 * 3 -1 and 6 * 3 + 1 are prime
      both 6 * 5 -1 and 6 * 5 + 1 are prime.


      Note: It's necessary to exclude 0 from the permitted values of both m
      and n because

      If m = 0, and n is not zero, then

      (3 * m + 1) * n + (3 * n + 1) * m = 1 * n + (3 * n + 1) * 0 = n, which
      represents all integers.


      Kermit Rose < kermit@... >
    Your message has been successfully submitted and would be delivered to recipients shortly.