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Mathematics professor at Cambridge
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A variant of Goldbach Conjecture
Sorry, that wasn't what I meant to ask. I asked the question I actually wanted to ask in a comment on Harald's answer above, which he has now answered. (The question was whether one could do it for all N and not just sufficiently large N.) 
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A variant of Goldbach Conjecture
Can your work can be adapted to prove the result for every $N$ when $p_1,p_2$ and $p_3$ are required to be less than $N$? 
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A variant of Goldbach Conjecture
You mean the result where you insist that $p_1$, $p_2$ and $p_3$ are less than $N$? 
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