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Re: [PrimeNumbers] Re:Goldbach: Probabilistic argument

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  • Kermit Rose
    Let k be an arbitrary positive integer 4. Let t be the number of primes strictly between 2 and k Let s be the number of primes strictly between k and 2k-2
    Message 1 of 1 , Dec 21, 2005
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      Let k be an arbitrary positive integer > 4.



      Let t be the number of primes strictly between 2 and k

      Let s be the number of primes strictly between k and 2k-2

      Probability that odd p < k is prime is t/(k-3)

      p < 2k - p < 2 k - 2

      p < k
      p + k < 2 k
      k < 2 k - p


      probability that 2k - p is prime is s/(k-3)



      Probability that p and 2k - p are prime is

      ( t/[ k-3 ] ) ( s/ [k - 3 ] ) = ts / [ k-3]^2

      So the expected number of prime pairs (p, 2k-p) is t s / [ k-3 ].




      Does this expected number of pairs increase or decrease as k increases?


      If it increases, as k increases, then the Goldbach conjecture is almost
      certainly true.

      If it decreases as k increases, then the Goldback conjecture is in doubt.

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