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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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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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