## A formula regarding Brun's sieve and the Goldbach problem

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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.
Message 1 of 1 , Jun 10, 2005
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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