A Sieve Method for Generating 1+1 Prime Pairs and Goldbach Function

A 2-step sieve method for generating 1+1 prime pairs, satisfying Goldbach's conjecture that every even integer greater than 2 can be written as the sum of two primes, has been disclosed. By direct counting based on indication of the Chinese remainder theorem, the so-called Goldbach function, G(2n), which is the number of distinct ways 2n can be expressed as the sum of two prime numbers, for a positive integer is generated accordingly. The result is G(2n) ~ (n-1)[(3-2)/(3-1)][(5-2)/(5-1)]… [(px-2)/(px-1)](1-1/2)(1-1/3)(1-1/5)… [1-1/(py)] x Fcg(n), where px ~ n1/2, py ~ (2n)1/2, and Fcg(2n) is a function defined as follows: Fcg(2n)=1, if there is no prime number p, 2 Fcg(2n+2) > 1, and, thus, G(2n) > G(2n+2) when n=papbpc…. The peak distribution can explain the so-called Goldbach's comet. Since n>>ln(2n) & Fcg(2n)>1 =>G(2n)>>1 for a large 2n even number, there exists at least a pair of primes such that 2n is the sum of these two primes. This is also true for every small 2n>2 by inspection or direct calculation without mathematical approximations. Therefore, every even integer greater than 2 can be written as the sum of two primes.