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.

  • $\begingroup$ For the "Theorem" you'd need at least $E>6$, since it's not true for $E=4,6$. $\endgroup$ – Zander Apr 24 '12 at 21:43
  • $\begingroup$ Right. I knew that and blew right by it in the post, which I've now corrected. $\endgroup$ – Keith Backman Apr 24 '12 at 22:36
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    $\begingroup$ How can you title this, "a different approach...," and then ask whether this approach has been pursued before? If you don't know whether the approach has been pursued before, please edit the word "different" out of the title. $\endgroup$ – Gerry Myerson Apr 25 '12 at 6:01
  • $\begingroup$ A more concise statement of the "conjecture" would be "Every even number $E\gt6$ that has a Goldbach representation has a Goldbach representation which contains at least one prime number which is not a member of $A$." By the way, "$\{A\}$" denotes the set containing a single object $A$; what you mean is just "$A$", the set called $A$. $\endgroup$ – joriki Apr 25 '12 at 6:09
  • $\begingroup$ My math dictionary defines 'theorem' as (1) A general conclusion proposed to be proved upon the basis of certain given hypotheses (assumptions). (2) A general conclusion which has been proved. $\endgroup$ – Keith Backman Apr 25 '12 at 14:43

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