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Every prime number except 2 and 3 has the form 6k+/-1. So far as has been established, sufficiently large even numbers can be represented as the sum of two primes in a plurality of ways. Consider 3 classes of even numbers: 6a-2; 6b; 6c+2. If it could be shown that (A) every sufficiently large even number of the form 6a-2 could be represented as the sum of two primes, each of the form 6k-1; and (B) every sufficiently large even number of the form 6b could be represented as the sum of one prime of the form 6k-1 and a second prime of the form 6k+1; and (C) every sufficiently large even number of the form 6c+2 could be represented as the sum of two primes, each of the form 6k+1; then Goldbach would be established. My question is: has anyone ever proved A, B, or C? Comment: Values of k for primes of the form 6k-1 are given in OIES listing A024898, and values of k for primes of the form 6k+1 are given in OIES listing A024899. Proof of A is tantamount to proving that any integer > 1 is the sum of two elements of A024898. Proof of B is tantamount to proving that any integer > 1 is the sum of one element of A024898 qne one element of A024899. Proof of C is tantamount to proving that any integer > 1 is the sum of two elements of A024899.

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  • $\begingroup$ It is usually a good idea to split a problem into subcases but.. the issue here is that we actually know less about primes in arithmetic progressions than we do generic primes. So it doesn't seem to help. $\endgroup$ – user58512 Feb 27 '13 at 20:53
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No, nobody has ever proved A, B, or C. In fact, no matter what arithmetic progression $Qk+R$ you choose, nobody has ever proved that all sufficiently large numbers of that form can be written as the sum of two primes. (It is generally believed that any proof that worked for one such arithmetic progression could be adapted to prove the full Goldbach conjecture.)

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