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Can all natural numbers ($n\ge 6$) be represented as the sum of three primes? With computer I checked up to $10000$, but couldn't prove it.

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This is the Goldbach conjecture. Good luck! – vgty6h7uij May 23 '12 at 19:02
See the Goldbach Conjecture and the weak Goldbach conjecture. – Eric Naslund May 23 '12 at 19:03
up vote 3 down vote accepted

It was proved by Vinogradov that every large enough odd integer is the sum of at most $3$ primes, and it seems essentially certain that apart from a few uninteresting small cases, every odd integer is the sum of $3$ primes.

Even integers are a different matter. To prove that every even integer $n$ is the sum of three primes, one would have to prove the Goldbach Conjecture, since one of the three primes must be $2$, and therefore $n-2$ must be the sum of two primes.

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From the second edition of The Hardy-Littlewood Method by Robert C. Vaughan, the Corollary to Theorem 3.4 on page 33 is that all sufficiently large odd number is the sum of three primes.

For even numbers and two primes, let $E(n)$ be the count of exceptions up to $n,$ meaning the count of even numbers that are not the sum of two primes. Let $A$ be a positive constant. The Corollary to Theorem 3.7 on page 36 is that there is a positive constant $C$ (where $C$ depends on $A$) such that $$ E(n) \leq C n (\log n)^{-A}. $$ So failures are eventually uncommon. I don't see that taking $p_1 + p_2 + 2$ improves matters very much.

Vaughan also refers to a book by one of the main investigators in this area, Additive Theory of Prime Numbers by L.-K. Hua (1965).

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Scientific American reports that Terence Tao is making progress on the weak Goldbach Conjecture. – Stefan Smith May 23 '12 at 23:23
Scientific American couldn't even write down correctly the WGC, so I'd take this declarations with some doubt. – DonAntonio May 24 '12 at 20:19
It doesn't much matter. If he announces a partial result, that is what counts for us. i always liked Scientific American as a kid. It was only much later that I read that they had a standing no-equations policy, which made considerable trouble for the occasional mathematician writing for them. I guess they would not need equations to talk about Goldbach, or perhaps they have weakened their stance. – Will Jagy May 24 '12 at 20:43

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