Open problem, Arithmetic and number theory, posed 1742

Goldbach conjecture

OpenLandau #1

Every even integer $n>2$ can be written as the sum of two primes: $n = p_1 + p_2$ .

Research frontier as of 2026

As of 2026 the (binary/strong) Goldbach conjecture is open. The strongest unconditional result toward it is still Chen's theorem (1973): every sufficiently large even number is $p + P_2$ , a prime plus a number with at most two prime factors. The ternary conjecture (every odd number $>5$ is a sum of three primes) is fully proved: Helfgott's 2012–2013 arXiv preprints (culminating in The ternary Goldbach conjecture is true, arXiv:1312.7748) are being consolidated into the forthcoming monograph The Ternary Goldbach Problem (Annals of Mathematics Studies, Princeton University Press), and the result is accepted as a completed theorem, though it does not imply the binary case. Computationally the binary conjecture has been verified for every even number up to $4\times10^{18}$ (Oliveira e Silva, Herzog, Pardi, 2014), with no counterexample found.

Best known results

Chen's theorem (1973): every sufficiently large even number is $p + P_2$ (a prime plus a product of at most two primes).
The ternary Goldbach conjecture is fully proved unconditionally for all odd $n>5$ (Helfgott, 2013, arXiv:1312.7748; forthcoming monograph in Annals of Mathematics Studies).
Verified by computer for every even number up to $4\times10^{18}$ , with no exception found (Oliveira e Silva, Herzog, Pardi, 2014).

Tools and where they stop

Tool	Achieved	Where it stops
Sieve methods (Chen's switching principle)	Gets every large even number down to $p + P_2$ , a near-miss with at most one extra prime factor.	The sieve-theoretic parity problem prevents distinguishing primes from semiprimes using these methods alone, so $P_2$ cannot be forced down to a single prime.
Circle method (Hardy–Littlewood / Vinogradov)	Fully resolves the three-prime (ternary) case unconditionally for all odd $n>5$ (Helfgott, 2013), by controlling major and minor arcs explicitly.	Matching a sum to exactly two primes needs much finer control of minor arcs than the method currently provides, so it does not reach the binary case.
Large-scale computer verification	Confirms no counterexample exists below $4\times10^{18}$ (Oliveira e Silva, Herzog, Pardi, 2014).	Can never prove a statement about all infinitely many even numbers, only rule out counterexamples below the checked bound.

Open questions

Can Chen's $p + P_2$ result ever be pushed down to two actual primes, or does the sieve-theoretic parity problem present a fundamental barrier requiring genuinely new ideas?
Could the major-arc control developed for the ternary case, combined with new sieve input, be adapted to make progress on the binary conjecture?

References

Chen Jingrun (1973). On the Representation of a Larger Even Integer as the Sum of a Prime and the Product of at Most Two Primes · DOI:10.1142/9789812776600_0021
Harald Andrés Helfgott (2013). The ternary Goldbach conjecture is true · arXiv:1312.7748 [preprint, not peer-reviewed]
Tomás Oliveira e Silva, Siegfried Herzog, Silvio Pardi (2014). Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot10^{18}$ · DOI:10.1090/S0025-5718-2013-02787-1