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?
Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements
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number-theory
combinatorics
intuition
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0[Crossposted to MO](http://mathoverflow.net/questions/74973/) – 2011-09-09
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0X is also infinite since there is infinite number of primes – 2011-09-09
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0anon,minimal element of X is k=1 so 2 can be represented as 2*1 – 2011-09-09
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0Dan,it states..."Let X be a set of natural numbers k_i....with the property..." – 2011-09-09
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0I've edited the question to clarify the meaning pedja had in mind. @Dan – 2011-09-09
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0Dan,I think that i was precise enough but I will accept any profound criticism of formulation – 2011-09-09
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0anon,you are right..this is better formulation – 2011-09-09
1 Answers
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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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0nice observation – 2011-09-08
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0Craig, I cannot see how to prove this using Goldbach. Any hints? – 2011-09-08
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0Consider 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 $Y$. – 2011-09-08
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0To clarify, if Goldbach's Conjecture is true, then this statement is true, because $6n$ is the sum of two primes of this form for $n>1$. – 2011-09-14
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0@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 – 2011-09-14
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0If 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