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  "body": "\n## Statement\n\nThere is an integer $N_0$ such that every odd integer $N \u003e N_0$ is a sum of three [[prime numbers|01826KMG2H]].\n\n## Remarks\n\nVinogradov proved the theorem in 1937 by the [[Hardy–Littlewood circle method|ZQFBEX62D0]].\nThe proof gives an asymptotic formula: for every $A \u003e 0$,\n$$\\sum_{p_1 + p_2 + p_3 = N} \\log p_1 \\log p_2 \\log p_3 = \\tfrac12 \\mathfrak{S}(N) N^2 + O\\!\\left(\\frac{N^2}{(\\log N)^{A}}\\right),$$\nwhere\n$$\\mathfrak{S}(N) = \\prod_{p \\mid N} \\left(1 - \\frac{1}{(p-1)^2}\\right) \\prod_{p \\nmid N} \\left(1 + \\frac{1}{(p-1)^3}\\right).$$\nFor odd $N$ the singular series satisfies $\\mathfrak{S}(N) \u003e 0.6$, so the main term dominates for large $N$; for even $N$ the factor at $p = 2$ is $0$.\nThe major arcs are handled by the [[Siegel–Walfisz theorem|XEPW251A97]], and the new step was Vinogradov's bound for the exponential sum $\\sum_{p \\le N} \\log p \\; e(p\\alpha)$ on the minor arcs.\nThe first proof gave no explicit $N_0$, because the Siegel–Walfisz constants are ineffective; later proofs made $N_0$ explicit.\nIn 2013 Harald Helfgott proved that every odd integer greater than $5$ is a sum of three primes.\n\n## Explanations\n\n### L1 human\n\nThe weak Goldbach conjecture says that every odd number greater than $5$ is a sum of three primes, as in $21 = 3 + 7 + 11$.\nVinogradov proved this for all odd numbers beyond some huge bound.\nHis method also counts, approximately, how many ways there are to do it.\n\n### L2 human\n\nThe theorem is the model application of the circle method to primes.\nThe case of three primes works because the minor arcs can be bounded with one factor estimated pointwise and the other two in mean square, using Parseval's identity.\nFor two primes, which is [[Goldbach's conjecture|063JNBSMJ0]], this argument fails: the mean square alone is as large as the expected main term.\nThe strongest results towards Goldbach's conjecture by other means are sieve results such as [[Chen's theorem|WE0RQTR68H]].\n",
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