# Is this equivalent to the Goldbach conjecture (modified slightly)?

Let $P \subset N$ be the set of primes. Is it true that the following statements are equivalent?

(1) There exists a natural number $N$ such that for all even natural numbers $n \geq N$, there exists two primes $p,q \in P$ such that $n=p+q$.

(2) There exists a finite set $S \subset N$ such that for all even $n \geq 4$ there exists $p,q \in P \cup S$ such that $n=p+q$.

(EDIT: One direction is easy. That is (1) implies (2). I have a feeling (2) implies (1) is possible to show. I'll admit I am somewhat confused how to show non-equivalence. I think that requires knowing the truth of the Goldbach conjecture as (2) implies (1) is only false if (2) is true and (1) is false. I'll accept a good argument why any attack using (2) to show (1) would likely fail in place of showing inequivalence if they are inequivalent.)

My motivation comes from thinking about sums of sets in general, that is $A+A$, and wondering just how important the distribution of the primes in the limit is or how much O(1) primes matter in making the Goldbach conjecture hard. My guess is that the limiting distribution is the most important and hence proving a statement like (2) is just as hard as proving the Goldbach conjecture.

• So it is theoretically possible that GC is false yet (2) above is true, hence the statement "(2) $\implies$ GC" is false... Oct 19, 2016 at 15:38
• @barakmanos, the only way a logical statement of the form $P\implies Q$ can be false is if $P$ is true and $Q$ is false. So to say that "(2) implies GC" is false is to say that (2) is true and GC is false. That's a very strong statement! Oct 19, 2016 at 15:53
• @BarryCipra: What about "true $\implies$ unknown"? Oct 19, 2016 at 15:56