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Let $X$ be the set of natural numbers $k_i$, $k_i \geq 1$, with the property that at least one of the equations $p_i = $6$ k_i \pm 1$ gives the $i$-th prime number (disregarding $2$ and $3$), and define the set $Y$ to be $Y = \mathbb{N} \setminus \{1\}$. Is it true that each element of Y can be represented as $2 k_i$ or as the sum $k_i + k_j$, where $k_i$ and $k_j$ are both elements of set X?

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    anon,you are right..this is better formulation2011-09-09

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This statement is implied by Goldbach's Conjecture, and does not look to be much easier to prove than the conjecture itself.

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    If this statement is false, then so is Goldbach's conjecture. However, it is possible (though unlikely) that this statement is true and Goldbach's conjecture is false. That is because this statement is slightly weaker than GC.2011-09-14