The binary (strong) Goldbach conjecture: is every even integer greater than 2 a sum of two primes?

This problem concerns what can be rigorously said about the possible existence of Goldbach counterexamples beyond any computationally verified range, and whether primality itself is essential or only distributional features of the summand set matter. The literal 'shift the primes by a fixed t' subquestion is exactly equivalent to the binary Goldbach conjecture, and the rigorous theory of 'quantitative statements about counterexamples' is the exceptional-set program, so the problem is the binary Goldbach conjecture in its standard formulation, together with its derived quantitative form: bounding the exceptional set $E(X)$ of even integers greater than 2 up to $X$ that are not sums of two primes (best published bounds: $E(X) \ll X^{1-\delta}$ by Montgomery–Vaughan 1975, improved to $E(X) < X^{0.72}$ by Pintz 2018 and $E(X) = O(X^{7/10})$ by Zhao 2025, all far from the expected $E(X) = 0$). The conjecture is verified computationally up to $4\cdot 10^{18}$ and remains open (Bhowmik and Grimmelt 2026).

Background

In a 1742 letter to Euler, Christian Goldbach proposed that every even integer greater than 2 can be written as the sum of two primes; this is the binary (or strong) Goldbach conjecture, and it is among the oldest open problems in number theory. The ternary (weak) version — every odd integer greater than 5 is the sum of three primes — was proved by Helfgott (2013), building on Vinogradov's 1937 work, but the binary version remains open.

The main analytic tool is the Hardy–Littlewood circle method (1923): one studies the generating function $f(\alpha)=\sum_{p\le N}e^{2\pi i \alpha p}$ over the primes and integrates $f(\alpha)^2$; regions near rationals with small denominator (major arcs) are handled by the distribution of primes in arithmetic progressions, while the remaining minor arcs require bounds for exponential sums over primes. The method yields a conjectured asymptotic formula: the number of representations of a large even $n$ as $p+q$ should be of order $n/(\log n)^2$ times a multiplicative constant (the singular series, a product over the odd primes dividing $n$ that corrects for local arithmetic biases). In particular, the conjecture predicts that failures do not merely thin out but cease entirely.

The rigorous quantitative substitute is the exceptional set: $E(X)$ counts the even integers $n$ with $2 < n \le X$ that are not sums of two primes. Estermann (1938) proved that almost all even integers are sums of two primes, i.e. $E(X)=o(X)$. Montgomery and Vaughan (1975) obtained the first power saving $E(X)\ll X^{1-\delta}$ for some $\delta>0$, and a series of refinements (Chen–Pan, Li, Lu) followed; Pintz (2018, preprint) proved $E(X)<X^{0.72}$, and Zhao (2025, preprint) announced $E(X)=O(X^{7/10})$ with an ineffective constant. None of these results excludes any individual counterexample, however large: they control only the counting function. On the computational side, Oliveira e Silva, Herzog, and Pardi (2014) verified the conjecture for all even integers up to $4\cdot 10^{18}$, which remains the published verification record; every even integer beyond that range is untouched by both computation and theory. The closest structural approximation is Chen's theorem (1973): every sufficiently large even integer is the sum of a prime and a $P_2$ (an integer with at most two prime factors); the parity problem in sieve theory — the inability of standard sieves to distinguish numbers with an odd from those with an even number of prime factors — is the recognized obstacle explaining why sieves stop at this statement.

A complementary perspective asks whether primality itself is essential for two-summand representability or whether only distributional features of the summand set matter. The rigorous content behind this question: what the circle method actually uses about the primes is their averaged distribution in arithmetic progressions (of Bombieri–Vinogradov type, i.e. equidistribution of primes among residue classes on average over moduli up to about $X^{1/2}$), so the density-type and exceptional-set conclusions hold for any hypothetical prime-like sequence with the same averaged distribution — such conclusions cannot distinguish the primes from a distributional surrogate. Conditional results sharpen this: Languasco and Perelli (1996) showed that the Generalized Riemann Hypothesis together with a generalized pair-correlation hypothesis for zeros of Dirichlet $L$-functions (a purely distributional assumption) yields strong exceptional-set bounds. At the ternary level, distributional pseudorandomness can genuinely suffice: Grimmelt (2022), via the transference circle method (a technique that transfers additive structure from a dense model set to a sparse pseudorandom one), proved that every sufficiently large $x \equiv 3 \pmod 4$ is a sum of three Fouvry–Iwaniec primes (primes of the form $a^{2}+p^{2}$ with $p$ itself prime, a sparse structured subset of the primes). On the other hand, the primes demonstrably deviate from naive random models at fine scales: Walker (2018) proved that the primes are not metric Poissonian, meaning that for almost every dilation factor $\lambda$ the fractional parts of $\lambda p$ fail to have Poisson-distributed pair correlations, so distributional heuristics cannot be assumed to capture prime behavior at the scales relevant to additive questions. For the binary problem, no hypothesis short of the full conjecture is known to suffice.

Problem Statement

Prove or disprove the binary (strong) Goldbach conjecture: every even integer greater than 2 can be written as the sum of two primes; equivalently, prove that the exceptional set $E(X) = \#\{n \le X : n \text{ even}, \; n > 2, \; n \text{ not a sum of two primes}\}$ vanishes identically (note $E(4\cdot 10^{18}) = 0$ is known by computation), or give a valid disproof of the conjecture (for instance by exhibiting a single explicit even integer $N > 4\cdot 10^{18}$ together with a verifiable certificate that $N - p$ is composite for every prime $p \le N/2$). The derived quantitative form of the problem — the pathway along which all unconditional partial results lie — is to bound $E(X)$: improve the known estimates $E(X) \ll X^{1-\delta}$ (Montgomery–Vaughan), $E(X) < X^{0.72}$ (Pintz), and $E(X) = O(X^{7/10})$ (Zhao) by driving the exponent down toward the conjectured truth $E(X) = 0$, a strict improvement over Zhao's exponent $7/10$ being the current published frontier. Translating both prime summands by a fixed integer $t$ reduces to the same conjecture, since $N = (p+t)+(q+t)$ if and only if $N - 2t = p+q$; any quantitative statement about counterexamples is exactly an exceptional-set bound on $E(X)$, since a deterministic set admits no literal probabilistic bound.

References

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2. Estermann, T. (1938). On Goldbach's problem: Proof that almost all even positive integers are sums of two primes. Proceedings of the London Mathematical Society, Second Series 44, 307–314. DOI: 10.1112/plms/s2-44.4.307. https://doi.org/10.1112/plms/s2-44.4.307
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14. Zhao, Genheng (2025). The exceptional set of Goldbach problem and Linnik's constant. Preprint, arXiv:2511.05631 [math.NT]. https://arxiv.org/abs/2511.05631
15. Bhowmik, Gautami; Grimmelt, Lasse (2026). The exceptional set of the Goldbach problem. Preprint, arXiv:2607.27282 [math.NT]. https://arxiv.org/abs/2607.27282