All primes are of the form 6*n+1 or 6*n-1. The sum of two primes must then be,
1 12*n-2
2 12*n or
3 12*n +2. Therefore, it si possible for all even numbers to be the sum of 2 primes.
The "conjecture' is not provable. The best I can do is to show that it's falsity is
highly improbable.
Let N be the even number in question. Let A_0 be next prime that is < N; B_0 is the next prime that is > N. We can add 3, 5, 7, 11, 13, 17, 19, 23, ... to A to get N.
What about adding 9? Let A_1 be the next lower prime. Let D_1 = A_0 - A-1, A simple
table shows how to add 9 to A_0.
D_1 ADD D_2 ADD D_3 DD
2 11 8 17 14 23
4 13 10 19 16
6 ?? 12 ??
8 17 14 23
10 19 16 ??
12 ?? 18 ??
14 23 20 29
Let's look at the bad D-sequences starting with D_1 = 6.
D
1 6
2 12
3 16
4 18
5
If D_1 = 6, then go to D_2 = A_0 - A_2