Hello,


Goldbach's conjecture states that every even integer greater than $3$ is the sum of two primes. I'm interested in a weaker assertion: has it been proven that every positive integer $n$ such that $6\vert n$ is the sum of two primes?
Thanks in advance.



Is every positive multiple of 6 the sum of two primes?
No. This would imply that every odd number at least $7$ is the sum of $3$ primes, since you can subtract $3$, $5$, or $7$ according to its residue mod 3. But that is not known. The strongest results known are that every sufficiently large odd number is the sum of $3$ primes, and that every odd number at least $11$ is the sum of $5$ primes.


