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

Diophantine equation and twin primes

Expand Messages
  • Sebastian Martin Ruiz
    ________________________________ Prove:  Theorem: Let c a positive even number 2 If (x,y)= (1,1) and (c^2/2-1,3c^2/2-2c-1) are the only positive integer
    Message 1 of 2 , May 26, 2013
    • 0 Attachment
      ________________________________



      Prove: 



      Theorem:

      Let c a positive even number >2

      If (x,y)= (1,1) and (c^2/2-1,3c^2/2-2c-1) are the only positive integer solutions

       of the polynomial 

      -3x^2+y^2-2xy-4cx+4cy+4=0

       then c +1 and c-1 are twin primes



      Sincerely

      Sebastián Martín Ruiz

      [Non-text portions of this message have been removed]
    • Mark
      ... Based on Warren s (corrected) simplification to your equation -3x^2 + y^2 -2xy - 4cx + 4cy + 4 = 0 namely r^2 - x^2 - c^2 + 1 = 0 Then (r-x)(r+x) =
      Message 2 of 2 , May 27, 2013
      • 0 Attachment
        --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:
        >
        > ________________________________
        >
        > Prove: 
        >
        > Theorem:
        >
        > Let c a positive even number >2
        >
        > If (x,y)= (1,1) and (c^2/2-1,3c^2/2-2c-1) are the only positive integer solutions
        >
        >  of the polynomial 
        >
        > -3x^2+y^2-2xy-4cx+4cy+4=0
        >
        >  then c +1 and c-1 are twin primes
        >
        >
        > Sincerely
        >
        > Sebastián Martín Ruiz
        >

        Based on Warren's (corrected) simplification to your equation

        -3x^2 + y^2 -2xy - 4cx + 4cy + 4 = 0

        namely

        r^2 - x^2 - c^2 + 1 = 0

        Then

        (r-x)(r+x) = (c-1)(c+1)

        If both c-1 and c+1 are prime, then clearly there are only two options:

        r-x = c-1 and r+x = c+1

        OR

        r-x = 1 and r+x = (c-1)(c+1)

        Follow each of those options and you easily arrive at the two results.

        (x,y) = (1,1) and ( (c^2)/2 - 1,3(c^2)/2 - 2c - 1)

        Mark
      Your message has been successfully submitted and would be delivered to recipients shortly.