= Vinogradov's three-primes theorem
{c}
{title2=$n=p_1+p_2+p_3\quad(n\text{ sufficiently large and odd})$}
Every sufficiently large odd integer is a sum of three primes, with repetitions allowed. The weighted count is asymptotic to $\mathfrak S(n)n^2/2$. <Siegel–Walfisz theorem> controls the <major arcs>, and Type I/II estimates for the prime <exponential sum> make the <minor arcs> negligible. Removing prime-power contributions gives genuine prime representations. This is the sufficiently-large theorem, not by itself a proof for every small odd integer.
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