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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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Crossposted to MO –  t.b. Sep 9 '11 at 5:35
    
X is also infinite since there is infinite number of primes –  pedja Sep 9 '11 at 5:53
    
anon,minimal element of X is k=1 so 2 can be represented as 2*1 –  pedja Sep 9 '11 at 6:00
    
Dan,it states..."Let X be a set of natural numbers k_i....with the property..." –  pedja Sep 9 '11 at 6:05
    
I've edited the question to clarify the meaning pedja had in mind. @Dan –  anon Sep 9 '11 at 6:14
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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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nice observation –  pedja Sep 8 '11 at 17:19
    
Craig, I cannot see how to prove this using Goldbach. Any hints? –  Srivatsan Sep 8 '11 at 17:33
    
Consider the numbers ${6n-2, 6n, 6n+2}$. If at least one of these can be expressed as the sum of two primes (which are not 3), then $n$ is in $X+X$. The not-3 bit is tricky, I'll admit, but that's a vanishingly small proportion of $Y$. –  Craig Sep 8 '11 at 17:49
    
To clarify, if Goldbach's Conjecture is true, then this statement is true, because $6n$ is the sum of two primes of this form for $n>1$. –  Craig Sep 14 '11 at 16:28
    
@Craig,If this statement isn't true than Goldbach's conjecture isn't true also or if one prove that this statement can't be proved or disproved than the same conclusion applies to Goldbach's conjecture –  pedja Sep 14 '11 at 17:14
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