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FW: The MAGIC KEY of primes PROOF

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  • Dimitris Valianatos
    From: dim-val@hotmail.com To: dim-val@hotmail.com; kon_val@hotmail.com Subject: The MAGIC KEY of primes PROOF Date: Sun, 24 Jun 2012 03:15:55 +0300 Hi Let the
    Message 1 of 1 , Jun 25, 2012
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      From: dim-val@...
      To: dim-val@...; kon_val@...
      Subject: The MAGIC KEY of primes PROOF
      Date: Sun, 24 Jun 2012 03:15:55 +0300


      Hi

      Let the
      pairs(e1, e2)wheree1, e2are even numbers, e2-e1 = 2
      andp = (e1 + e2) / 2is a primenumberamong
      them?
      andapply therulein
      the followingtriads?
      (4,5,6), (6,7,8), (​​10,11,12), (12,13,14),(16,17,18), (18,19,20), (22,23,24), (​​28,29,30),
      (30,31,32),(36,37,38), (40,41,42), (42,43,44), (​​46,47,48),
      ..., (e1, p, e2)
      The rule is:
      ifp = 3
      mod 4(ore1 = 2
      mod 4)then  we formthe
      productΠe2/e1
      ifp
      = 1 mod 4(ore1 = 0 mod 4) then  we formthe productΠe1/e2
       
      The product of:     
      (4/6) * (8/6) * (12/10) * (12/14) *
      (16/18) * (20/18) * (24/22) * (28/30) * (32/30) * (36/38) *
      (40/42) * (44/42) * (48/49)* ... = 1
      for each p prime and  p>
      3, p -> {5,7,11,13,17,19,23, ..}
       
      PROOF:
       
      Π(1 -X(Pk)/Pk)
      = 4/pi for k=2 to infinite, Pk is kth prime.
      Π(1+X(Pk)/Pk)
      = 2/pi for k=2 to infinite, Pk is kth prime.
       
      Where:
              { +1  if P=1 mod 4
      X(P)={
                  { -1   if P=3 mod 4
       
      For k>=3 the 2
      products are equals.
      Π(1 -X(Pk)/Pk) = 3/pi
      Π(1+X(Pk)/Pk) = 3/pi
       
      Because for k=2 we
      have (1+1/3)=4/3 and (1-1/3)=2/3
      So 4/pi*3/4=3/pi
      and 2/pi*3/2=3/pi
       
      The quotient
      (Π(1 -X(Pk)/Pk)) / (Π(1+X(Pk)/Pk)) = ((1-1/5)/(1+1/5))*((1+1/7)/(1-1/7))*…=
      =(4/5)/(6/5)*(8/7)/(6/7)*… = 4/6*8/7*12/10*… = (3/pi)/(3/pi)=1  for k>=3
       
      (I call this the MAGIC KEY of primes.)

      Best regards
      Dimitris Valianatos 

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