The binary Goldbach conjecture asserts that every even integer greater than
4 is the sum of two primes. In this paper, we prove that there exists an
integer Kα such that every even integer x>pk2 can be expressed as
the sum of two primes, where pk is the kth prime number and k>Kα. To prove this statement, we begin by introducing a type of double
sieve of Eratosthenes as follows. Given a positive even integer x>4, we
sift from [1,x] all those elements that are congruents to 0 modulo p or
congruents to x modulo p, where p is a prime less than x.
Therefore, any integer in the interval [x,x] that remains unsifted is
a prime q for which either x−q=1 or x−q is also a prime. Then, we
introduce a new way of formulating a sieve, which we call the sequence of
k-tuples of remainders. By means of this tool, we prove that there exists an
integer Kα>5 such that pk/2 is a lower bound for the sifting
function of this sieve, for every even number x that satisfies pk2<x<pk+12, where k>Kα, which implies that x>pk2(k>Kα) can be expressed as the sum of two primes.Comment: 32 pages. The manuscript was edited for proper English language by
one editor at American Journal Experts (Certificate Verification Key:
C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made in
sections 1, 6, 7 and