The ternary Goldbach theorem (Vinogradov 1937, Helfgott 2013)

The proof follows the circle method exactly as sketched earlier in this branch of number theory, applied to the von Mangoldt-weighted exponential sum $F(\alpha)=\sum_{p\le n}\log p\; e(p\alpha)$, with representation count weighted by $\int_0^1 F(\alpha)^3 e(-n\alpha)\,d\alpha$.

On the major arcs (short intervals around rationals $a/q$ with $q$ small), the Siegel–Walfisz theorem on primes in arithmetic progressions — uniform for moduli $q$ up to any fixed power of $\log n$ — lets one evaluate the integral there explicitly, producing a main term $\tfrac12\mathfrak S(n)\,n^2$ where $\mathfrak S(n)$, the "singular series," is a product of local densities, one for each prime, measuring how often $n$ is solvable as $p_1+p_2+p_3$ modulo that prime. For odd $n$, every local condition is solvable (there is no obstruction analogous to the parity obstruction that plagues the binary case), so $\mathfrak S(n)$ is bounded away from $0$, and the main term is genuinely positive and of size $n^2$.

On the minor arcs, Vinogradov's exponential-sum estimate for primes — a highly nontrivial bound $|F(\alpha)|\ll n(\log n)^4/q^{1/2}$ valid there — shows the contribution is $o(n^2)$, strictly smaller than the main term, so it cannot cancel the positive contribution from the major arcs. Combining the two gives a representation count that is positive once $n$ is large enough, which is Vinogradov's 1937 theorem.

Helfgott's 2013 completion made every step above fully explicit rather than merely "sufficiently large": sharper major-arc and minor-arc estimates (including explicit, computer-verified bounds on zeros of Dirichlet $L$-functions up to a specific height) closed the gap between the explicit threshold where the analytic argument works and the range that had already been checked by direct computer search, yielding an unconditional proof for literally every odd $n>5$.