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A formula regarding Brun's sieve and the Goldbach problem

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  • antonioveloz2
    While studying Brun s sieve from his paper The Sieve of Eratosthenes and the Theorem of Goldbach , I thought of some changes that could be done to the sieve.
    Message 1 of 1 , Jun 10, 2005
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      While studying Brun's sieve from his paper "The Sieve of Eratosthenes
      and the Theorem of Goldbach", I thought of some changes that could be
      done to the sieve. I believe the formula, or at least a similar one
      must known, but in case it is not known I wanted to post it here for
      your scrutiny and so that I will have claim to it. I will post the
      formula and show some numerical evidence as to its accuracy and if
      anyone is interested I will go into more detail.

      Let FLOOR[x] be the greatest integer less than or equal to x.
      Let mu[n] be the mobius function found at
      http://mathworld.wolfram.com/MoebiusFunction.html

      Let w[n] be the number of distinct prime factors of the integer n.
      Let R[n] be the number of ways of writing n as the sum of two primes,
      where n=p1+p2 and n=p2+p1 are counted as two representations.

      Let (n,k) be the greatest common divisor of n and k.
      Let Phi[n] be the euler totient function and Pi[n] be the prime
      counting function.

      Then we have the following lower bound for R[n]

      R[n] >= 2*Pi[n] - 2*w[n] - phi[n] +
      (.5)* SUM(2<= j <= n/4 and (n,j)=1){Mu[n]*(2^(w[n]-1)-1)*FLOOR[(n/j)-
      4]}


      As for the accuracy of the formula…consider the following results

      24=8*3
      R[24] = 6
      My estimate is 6.

      The formula is exact or off by 1 for the following even numbers <=
      200;
      6,8,12,14,18,20,24,30,32,38,42,48,54,60,72,80,84,90,108,110,132,138,14
      0,150,180,

      but after that it is only exact for 770 and no other numbers <= 1000!

      26=2*13
      R[26]=5
      My estimate is 2.

      106=2*53
      R[106]=10
      My estimate is 0.

      The previous example shows that my method in GENERAL will NOT imply
      the Goldbach Conjecture, but it does provide partitions for certain
      KINDS of numbers depending on the prime factorization of the number.
      Consider the following examples

      2178=2*9*121
      R[2178]=136
      My estimate is 136

      2180=4*9*109
      R[2180]=82
      My estimate is 80

      2182=2*1091
      R[2182]=66
      My estimate is –19!

      2184=8*3*7*13
      R[2184]=170
      My estimate is 157

      2186=2*1093
      R[2186]=66
      My estimate is –19

      As I have said before my method will not imply Goldbach in general
      but in the future will speak more about my claim that for certain
      types of numbers my formula guarantees the existence of a goldbachian
      partition if anyone is interested.
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