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Reflection on the conjecture

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  • Kermit Rose
    Reflection on the conjecture that if q is the next prime after p, where p is a positive integer prime, and if q - p 2, then sqrt( [ p^2 + q^2 ] /2 -1 ) is
    Message 1 of 1 , Jul 6, 2006
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      Reflection on the conjecture

      that if q is the next prime after p,

      where p is a positive integer prime,

      and if q - p > 2,

      then

      sqrt( [ p^2 + q^2 ] /2 -1 ) is irrational.

      Case 1: q - p = 4.

      Then set

      p = 6 D + 1

      q = 6 D + 5

      p^2 + q^2 = [ 36 D^2 + 12 D + 1 ] + [36 D^2 + 60 D + 25 ]
      = 72 D^2 + 72 D + 26

      [p^2 + q^2 ]/2 = 36 D^2 + 36 D + 13

      [p^2 + q^2 ]/2 - 1 = 36 D^2 + 36 D + 12
      = 12 [ 3 D^2 + 3 D + 1 ]

      = 4( 3 [ 3 D^2 + 3 D + 1) )

      To be a counter example to the conjecture,

      z2 = 4(3 [ 3 D^2 + 3 D + 1] ) would have to be a perfect square,

      which is impossible,

      because z2 is a multiple of 3, but not a multiple of 9.


      Case q - p = 6

      Set p = 6 D + t, where t = either 1 or -1.


      Then q = 6(D+1) + t

      p^2 + q^2 = [36 D^2 + 12 t D + t^2] + [36 D^2 + 72 D + 36 + 12 t D + 12 t +
      t^2 ]

      = 72 D^2 + [ 24 t +72 ]D + [12 t + 2 t^2 ]

      [ p^2 + q^2]/2 = 36 D^2 + [ 12 t + 36 ] D + [6 t + t^2 ]

      = 36 D^2 + [12 t + 36 ] D + [6 t + 1 ]


      [ p^2 + q^2 ] /2 - 1 = 36 D^2 + [12 t + 36 ] D + 6 t

      To be a counter example to the conjecture,

      z2 = 36 D^2 + [ 12 t + 36 ] D + 6 t would need to be a perfect square.


      Case t = -1.

      z2 = 36 D^2 + 24 D - 6 = 3 ( 12 D^2 + 8 D - 2)

      cannot be a perfect square because

      z2 is divisible by 2 and not divisible by 4.

      Case t = 1

      z2 = 36 D^2 + 48 D + 6 = 6(6 D^2 + 8 D + 1)

      cannot be a perfect square because

      z2 is divisible by 6 and not divisible by 36.


      case q - p = 6 M with M > 0.

      p = 6 D + t with t^2 = 1.

      q = 6 D + 6 M + t = 6 (D + M) + t

      p^2 + q^2 = 36 D^2 + 12 t D + t^2
      +36 D^2 + 72 M D + 36 M^2
      + 12 t D + 12 t M
      + t^2

      z2 = [p^2 + q^2]/2 - 1 = 36 D^2 + 24 t D + 72 MD + 18 M^2 + 6 t M

      = 3 (12 D^2 + 8 t D + 24 M D + 6 M^2 + 2t M)

      Perhaps this case can lead to a counter example.

      It's not clear to me yet.
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