The Problem and Its Status

Status as of July 2026. The binary Goldbach conjecture remains open: it is neither proved nor refuted. Every even integer up to $4 \cdot 10^{18}$ is verified to be a sum of two primes (Oliveira e Silva, Herzog, Pardi, Math. Comp. 83, 2014) — an empirical fact, not a proof. The strongest unconditional partial result remains of Chen type: every sufficiently large even $N$ is the sum of a prime and a number with at most two prime factors (Chen, 1973). The best published exceptional-set bound is $E(X) \ll X^{0.879}$ (Lu, 2010); stronger bounds exist as unreviewed preprints and are labeled as such below.

What this site does not claim

No proof of the conjecture, and no claim of being close to one. The results of this project are partial: corollaries derived from published work, one theorem conditional on the (widely disbelieved) Siegel-zero hypothesis, and machine-checked combinatorial facts — none of which advances the conjecture itself. The approaches studied here map the known impossibility frontier — the parity barrier — rather than bypass it; every route that once seemed to bypass it has been refuted, and the refutations are published on this site.

Statement

On 7 June 1742, Christian Goldbach, then working in Moscow, wrote a letter to Leonhard Euler. In the ensuing correspondence the following assertion took shape; it has remained open for almost three centuries.

Binary (strong) Goldbach conjecture. Every even integer $N > 2$ is the sum of two primes:

The conjecture is unproved and unrefuted. Everything listed on this page is a partial result, an approximation, or a conditional theorem; none of it constitutes a proof.

The natural quantitative measure of the problem is the representation counting function

The conjecture is equivalent to the assertion that $r(N) \ge 1$ for every even $N \ge 4$. The Hardy–Littlewood circle method predicts an asymptotic formula for $r(N)$ whose leading factor is the so-called singular series:

The Hardy–Littlewood heuristic asserts that $r(N) \approx \mathfrak{S}(N)\,\frac{N}{(\ln N)^{2}}$. Note that $\mathfrak{S}(N) > 0$ for even $N$: the heuristic predicts an abundance of representations. Turning this heuristic into a proof is precisely what no one has achieved so far.

Computational verification

Direct computation confirms the conjecture to a remarkable depth. The record is due to Oliveira e Silva, Herzog, and Pardi: verified up to $4 \cdot 10^{18}$

T. Oliveira e Silva, S. Herzog, S. Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$”, Mathematics of Computation 83 (2014), 2033–2060. Every even integer up to $4 \cdot 10^{18}$ is indeed a sum of two primes. Published article: doi:10.1090/S0025-5718-2013-02787-1; the project page describing the computation, formerly at sweet.ua.pt, is preserved in the Internet Archive.

Computational verification is not a proof: it covers a finite — albeit enormous — range and says nothing about integers beyond it.

Chen’s theorem and the “(1 + a)” scale

The principal unconditional result of the twentieth century in the direction of Goldbach is due to Chen Jingrun (announced 1966, full proof published 1973): Theorem

Chen’s theorem. Every sufficiently large even integer $N$ has a representation

where $p$ is prime and $P_{2}$ is a product of at most two primes (an “almost prime”).

Progress is conveniently measured on the “(1 + a)” scale: the assertion (1 + a) means a representation $N = p + P_{a}$, where the second summand has at most $a$ prime factors.

For half a century, Chen’s theorem remained unsurpassed on this scale.

The current frontier: a claimed (1 + 1.9)

Li & Liu, preprint arXiv:2606.05224 (2026): preprint v1, not peer-reviewed

The result claimed in the preprint: every sufficiently large even integer $N$ has representations of the form

where $p$ and $q$ are prime. The case $r = 1$ is Goldbach itself; the claim, however, allows a “correction factor” $r$ bounded by $q^{0.9}$. This would advance beyond Chen’s (1 + 2), but it is not a proof of the Goldbach conjecture: representations with $r = 1$ are neither isolated nor guaranteed.

Status: not refereed. The statement above is reported from version v1 of the preprint. Until independent review is complete, it should be regarded as preliminary, not as an established theorem.

The exceptional set

Another approach to the problem is to estimate how many even integers could be counterexamples at all. Denote by $E(X)$ the number of even $N \le X$ that are not a sum of two primes. The Goldbach conjecture is equivalent to $E(X) = 0$ for all $X \ge 4$; proved results give only power-saving upper bounds.

On ineffectivity. In the bound $E(X) = O(X^{7/10})$, the constant implicit in the symbol $O$ is ineffective: the proof provides no way to compute it. Consequently, this bound yields no explicit threshold beyond which counterexamples become “almost absent”, and a fortiori it cannot reduce the conjecture to a finite check.

Even with $E(X) = 0$ for all $X \le 4\cdot 10^{18}$ and power-saving bounds above, the exceptional set remains, strictly speaking, potentially non-empty.

Zeros of the zeta function and density estimates

Progress in additive prime number theory is closely tied to the distribution of zeros of the Riemann zeta function. One writes $N(\sigma, T)$ for the number of zeros $\rho = \beta + i\gamma$ with $\beta \ge \sigma$ and $|\gamma| \le T$.

Guth & Maynard — a new density estimate: Theorem

Preprint arXiv:2405.20552; published: Annals of Mathematics 203 (2026), no. 2, 623–675. The proved bound is

This strengthens the classical density estimates for zeros near the line $\Re s = 1$. Such estimates are a key ingredient of modern work on primes in short intervals and, indirectly, on binary additive problems.

By itself, this estimate does not prove the Goldbach conjecture; it shifts the technical foundation on which attacks on the conjecture are built.

Conditional “doors” to Goldbach

There are precise conditional implications: from certain (themselves unproved) hypotheses on the distribution of primes, binary Goldbach already follows. The cleanest known formulation is due to Huang & Li, preprint arXiv:2005.03811 (2020): Conditional result

If the Elliott–Halberstam conjecture and its Möbius-twisted variant hold with distribution levels whose sum exceeds $1$, then the binary Goldbach conjecture follows for all sufficiently large even integers.

This is a conditional result: both premises remain unproved. The best unconditional distribution levels (the Bombieri–Vinogradov theorem gives level $1/2$) do not yet add up to the required sum.

What remains

To summarize. The conjecture has been verified up to $4\cdot 10^{18}$, but verification is not proof. Chen’s theorem gives (1 + 2); the Li–Liu preprint claims (1 + 1.9); the case (1 + 1) has not been reached. Exceptional-set estimates show that counterexamples are “few”, not that there are none — and the strongest of them carries an ineffective constant. Conditional results turn Goldbach into a consequence of other unproved hypotheses: the door is marked, but the keys are still missing.

The fundamental reason the sieve stalls one step short of the goal is the parity barrier: classical sieve methods cannot in principle distinguish integers with an even number of prime factors from those with an odd number. A detailed discussion of this and other obstacles is given in a separate section: The parity barrier.