EVERY SUFFICIENTLY LARGE EVEN NUMBER IS THE SUM OF TWO PRIMES\\(with an alternative proof of Lemma 7.1)

With an alternative proof of the crucial Lemma 7.1, 36 pages.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 > 4 such that every even integer x > p 2 k can be expressed as the sum of two primes, where p k 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 > 4 such that p k /2 is a lower bound for the sifting function of this sieve, for every even number x that satisfies p 2 k Kα, which implies that x > p 2 k (k > Kα) can be expressed as the sum of two primes. In this version of the paper we give an alternative proof of the crucial lemma that allowed us to obtain this result