Consider the set A of prime numbers $p_i$ such that $p_i+6$ is not prime (listed in OEIS 140555; see comments thereto). Let 'Goldbach representation' mean a pair of odd prime numbers which sum to a given even number.
Theorem Conjecture: For any even number $E>6$ which has a Goldbach representation consisting of two members of set A, there is at least one other Goldbach representation of $E$ which contains at least one prime number which is not a member of set A.
My questions are: can anyone prove this
theorem conjecture? Or suggest how to attack it? Has this approach to Goldbach been pursued before?
Comments: If Goldbach is false, then there is an integer $K$ such that $2K$ is the smallest number that has no Goldbach representation. The following argument is true for all appropriate values of $a$ (and the
theorem conjecture could be restated for other values of $a$, using sets defined as $p_i$ such that $p_i+2a$ is not a prime), but I will discuss the specific instance of $a=3$. Since $N=2(K-3) < 2K, N$ has a Goldbach representation. However, if either of the primes in any Goldbach representation of $N$ can be supplemented by 6 to yield another prime, then $2K$ itself will have a Goldbach representation, which is counter to our assumption. So if Goldbach is false, all Goldbach representations of $2(K-3)$ must contain only members of A. Hence, proving the theorem conjecture stated is equivalent to proving Goldbach. Note that even if the theorem conjecture as stated (i.e. for $a=3$) is not true, the restatements of the theorem conjecture for every appropriate value of $a$ must also fail in order for the Goldbach conjecture to be false. Thus the Goldbach conjecture is equivalent to saying that there is some value for $a$ for which a theorem conjecture of the form given is true.