Vinogradov's three-primes theorem ($n=p_1+p_2+p_3(n\text{ sufficiently large and odd})$)

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.