# Sum of three primes

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. Also, are you sure you wanted to tag this as homework? –  Eric Naslund May 23 '12 at 19:03
+1 for the teacher that gives such homework! –  dtldarek May 23 '12 at 21:41

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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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.