Every sufficiently large even number is the sum of two primes
Abstract
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
K
α
such that every even integer
x>
p
2
k
can be expressed as the sum of two primes, where
p
k
is the
k
th prime number and
k>
K
α
. 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
K
α
>5
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
<x<
p
2
k+1
, where
k>
K
α
, which implies that
x>
p
2
k
(k>
K
α
)
can be expressed as the sum of two primes.