Goldbach's conjecture

This is the closest anyone has come to proving binary Goldbach with a fully rigorous, unconditional argument: it relaxes 'prime' to 'prime or semiprime' on one side, which sieve methods can handle even though they cannot isolate primes exactly.

The proof is a weighted sieve argument. Fix large even $n$ and consider, for each prime $p\le n$, whether $n-p$ has few prime factors. A naive sieve (Selberg's upper-bound sieve) can show that the count of $p\le n$ for which $n-p$ has at most two prime factors is not too small — but a plain sieve alone cannot distinguish "$n-p$ prime" from "$n-p$ has $3,4,\dots$ prime factors" strongly enough to conclude anything.

Chen's key device, the weighted (or "switching principle") sieve, assigns each candidate $p$ a weight built from two different sieve levels: an upper-bound sieve for the "bad" event that $n-p$ has three or more prime factors below a threshold $z\approx n^{1/3}$, and a lower-bound sieve for the total count of $p\le n$ with $n-p$ coprime to all primes below $z$. Subtracting a suitably weighted multiple of the first from the second produces a combinatorial sum that is provably positive once $n$ is large — showing that some $p$ survives with $n-p$ having at most $2$ prime factors above the threshold, i.e. $n-p$ is prime or a semiprime.

Making "provably positive" work requires estimating a bilinear form coming from the sieve remainder terms, using results of Bombieri–Vinogradov type on the distribution of primes in arithmetic progressions to large moduli (on average over moduli up to about $n^{1/2-\varepsilon}$); this is the deepest analytic input, and it is why the theorem needs $n$ "sufficiently large" rather than holding for all $n$ from the start.

The theorem stops at "prime or semiprime" and not "prime or prime" because sieve methods suffer from the parity problem: standard sieve weights cannot, by their construction, tell numbers with an even number of prime factors apart from numbers with an odd number, so they can never sieve down to primes alone from this kind of argument — semiprime is the sharpest target sieve theory can currently reach.

This is the deepest unconditional success of the circle method applied directly to the primes, and it turns the binary Goldbach conjecture from a completely open problem into one where at least the odd-sum analogue is fully settled.

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 $n>5$.