How to read this page. This is an honest map of what is and is not known about Goldbach's problem: not a proof. Throughout, p, q denote prime numbers; ℕ the positive integers, ℤ the integers, ℝ the reals, ℂ the complex numbers. π(x) is the prime-counting function (the number of primes ≤ x), and by the Prime Number Theorem π(x) ≈ x / ln x. We write log for the natural logarithm. The symbol r(n) denotes the number of ways to write n as an ordered sum of two primes. Two forms of the conjecture recur: the strong (or binary) form, about even numbers and two primes, which is OPEN; and the weak (or ternary) form, about odd numbers and three primes, which is PROVEN. Keep them strictly separate: nearly every confusion about "Goldbach being solved" comes from conflating the two.
0 · The claim, in one line
Every even integer greater than 2 is a sum of two primes. For instance 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. No one has ever found an even number that fails. No one has ever proven none can.
1 · The object: what it is and why anyone cares
The primes (2, 3, 5, 7, 11, 13, …) are defined by multiplication: a prime is an integer > 1 divisible only by 1 and itself. They are the multiplicative atoms of the integers; the Fundamental Theorem of Arithmetic says every integer factors uniquely into them. Goldbach's conjecture asks a question that crosses the grain of that definition: it asks how the primes behave under addition.
That crossing is the entire point, and the entire difficulty. Multiplication is where primes live natively; addition is a foreign country. There is no reason, buried in the definition of "prime," that adding two of them should land on every even target. The conjecture asserts a dense, universal additive regularity in a set defined by a multiplicative irregularity. The primes are not supposed to be this cooperative.
Chase the "why" to its root and you reach the deepest open theme in number theory: the primes behave, statistically, like a random set of the correct density, yet they are fully deterministic. The Prime Number Theorem says a number near n is prime with "probability" about 1 / log n. If the primes were genuinely a random set of that density, a short heuristic (Section 3) predicts that an even n should have on the order of n / (log n)² representations as p + q: an enormous number, growing without bound. Goldbach should not merely be true; it should be true with room to spare. The conjecture is hard not because it is implausible but because turning "the primes act random" into "therefore every single even number is hit" requires controlling the primes one target at a time, and our tools control them only on average.
So the real object of study is the tension between pseudorandomness and rigidity in the primes. Goldbach is the cleanest, most elementary-sounding place that tension shows up: a child can understand the statement, and no living mathematician can prove it.
2 · The precise statement / the mathematical setup
The two conjectures, stated exactly
Strong (binary) Goldbach: ∀ n ∈ ℤ, n even, n > 2 ⟹ ∃ primes p, q with n = p + q.Every even number above 2 is a sum of two primes. This is the OPEN problem.
Weak (ternary) Goldbach: ∀ n ∈ ℤ, n odd, n > 5 ⟹ ∃ primes p, q, r with n = p + q + r.Every odd number above 5 is a sum of three primes. PROVEN (Helfgott, 2013).
The strong form implies the weak form in one line: given odd n > 5, the number n − 3 is even and > 2, so if it is p + q then n = 3 + p + q. The converse does not hold: which is exactly why the weak form could be settled while the strong form stands. Weak solved does not mean strong solved.
A note on the original 1742 wording
In his letter to Leonhard Euler dated 7 June 1742, Christian Goldbach (using the then-common convention that 1 counts as prime) conjectured that every integer > 2 is a sum of three primes. Euler replied (30 June 1742) that this is equivalent to the sharper claim he attributed to Goldbach's earlier remark: every even integer ≥ 4 is a sum of two primes, which Euler said he regarded as certain though he could not prove it. The modern "two primes, even numbers" statement is thus properly the Euler form of Goldbach's conjecture. (With the modern convention that 1 is not prime, the small cases are handled by taking 2 as the one even prime.)
The counting function and the generating object
Define the representation count
r(n) = #{ (p, q) : p, q prime, p + q = n }.The number of ordered ways to write n as a sum of two primes. Goldbach (strong) ⟺ r(n) ≥ 1 for every even n > 2.
The analytic machinery studies r(n) through the exponential sum over primes. Writing e(θ) = e2πiθ and
S(α) = ∑p ≤ N e(pα),a "prime-detecting" wave; its size at a given frequency α records how the primes line up against that frequency.
one has the exact identity, for the number of prime pairs summing to n,
r(n) = ∫01 S(α)² e(−nα) dα.Squaring S(α) builds sums p+q; the integral picks out exactly those equal to n. The whole problem is: can this integral be forced to be positive for every even n?
This is the Hardy–Littlewood circle method, and everything serious that has been proven about Goldbach flows through analyzing that integral.
3 · Every connection: what it touches, what hangs on it, what it unlocks
The Hardy–Littlewood asymptotic (the quantitative heart)
In their 1923 paper Some problems of 'Partitio Numerorum' III, G. H. Hardy and J. E. Littlewood used the circle method to conjecture a precise formula for how many representations an even n should have:
r(n) ∼ 2 C₂ · (n / (log n)²) · ∏p | n, p > 2 ((p − 1)/(p − 2)), where C₂ = ∏p > 2 (1 − 1/(p−1)²) ≈ 0.6601…The predicted number of Goldbach representations of n. C₂ is the "twin-prime constant." It grows like n/(log n)², so large even numbers should have vastly many representations.
This is still a conjecture, but it is the shape every heuristic agrees on, and numerical data match it strikingly (the famous "Goldbach comet" scatterplot of r(n)). It tells us the strong conjecture is not a knife-edge: if it is ever false, it fails for a single small-ish number against overwhelming statistical pressure, not because representations run out at infinity.
What is logically tied to Goldbach
- The weak conjecture: implied by the strong one, now independently proven.
- The twin-prime constant C₂: the same constant governs the conjectured count of twin primes. Goldbach and the Twin Prime Conjecture are two faces of the same Hardy–Littlewood k-tuple philosophy about additive patterns in primes; progress on the underlying "primes are pseudorandom" problem would move both.
- Schnirelmann's theorem and additive bases: Goldbach is the extreme case (order 2) of the question "how many primes must you add to represent every integer?" This spawned the theory of additive bases and Schnirelmann density.
- Generalized Riemann Hypothesis (GRH): GRH does not prove strong Goldbach, but it sharpens the analytic tools decisively. Crucially, GRH does let one prove the weak conjecture for all odd numbers down to a checkable range (Deshouillers, Effinger, te Riele, Zinoviev, 1997, proved weak Goldbach conditionally on GRH): a stepping stone eventually made unconditional by Helfgott.
Honest scope note: Goldbach is not known to imply, or be implied by, the Riemann Hypothesis. It does not "unlock" RH and RH does not settle it. Claims that a proof of Goldbach would collapse number theory overstate the linkage: what a proof would really deliver is a genuinely new method for extracting pointwise (every-single-n) conclusions from average information about primes. That method is the prize.
4 · Every serious attack: and exactly how far each got
Brun's sieve (1915–1920)
Viggo Brun invented modern sieve theory expressly to attack Goldbach and twin primes. Around 1919 he proved that every sufficiently large even number is a sum of two numbers each having at most nine prime factors (the "9 + 9" theorem). This was the first genuine foothold: not primes, but numbers close to prime. The program of driving those factor counts down toward 1 + 1 dominated the next half-century.
The circle method and the ternary breakthrough: Hardy, Littlewood, Vinogradov (1923–1937)
Hardy and Littlewood (1923) set up the integral of Section 2 and, assuming GRH, proved the weak conjecture for all sufficiently large odd numbers. In 1937 Ivan Vinogradov removed the GRH assumption entirely, using his new method for estimating sums over primes on the "minor arcs." His result (unconditional ternary Goldbach for all sufficiently large odd n) is one of the landmarks of 20th-century number theory.
Vinogradov (1937): every sufficiently large odd integer is a sum of three primes.Unconditional. But "sufficiently large" meant a bound astronomically beyond any possible computer check.
The catch was effectiveness. Vinogradov's threshold was later made explicit: e.g. K. Borozdkin in the 1950s gave a bound around ee16.038, roughly 10106.6: hopelessly large to close by direct computation.
Chen's theorem (1966 announced, 1973 published)
Chen Jingrun pushed the sieve to its near-limit for the binary problem:
Chen (1973): every sufficiently large even n is p + P₂, where p is prime and P₂ is a product of at most two primes."1 + 2." The closest anyone has come to Goldbach itself: one summand prime, the other prime or a semiprime.
This is the current high-water mark on the structure of the conjecture. It is tantalizingly close: and, as Section 5 explains, the gap from "1 + 2" to "1 + 1" is not a small final step but a wall the method cannot cross.
The exceptional set: Estermann, Chudakov, van der Corput, Montgomery–Vaughan (1930s–1975)
If one cannot prove every even number works, one can bound how many could fail. Let E(x) be the number of even n ≤ x that are not sums of two primes. In the late 1930s Estermann, Chudakov, and van der Corput independently showed E(x) is small; in 1975 H. L. Montgomery and R. C. Vaughan proved a power saving:
E(x) ≪ x1 − δ for some fixed δ > 0.The even numbers that fail Goldbach (if any) are extraordinarily sparse: a density-zero set, in fact fewer than x^(1−δ) up to x.
So almost every even number is provably a sum of two primes. This is strong evidence and a real theorem: but "almost every" is precisely what the conjecture forbids us to settle for. A single exception, infinitely often or even once, would be consistent with every exceptional-set bound ever proven.
Schnirelmann and bounded sums (1930s onward)
Lev Schnirelmann (c. 1930) proved every integer > 1 is a sum of a bounded number of primes: originally a few hundred thousand, later driven down. Olivier Ramaré proved in 1995 that every even number is a sum of at most six primes; after Helfgott's ternary result it follows that every even number ≥ 4 is a sum of at most four primes. Still not two.
The weak conjecture, finished: Helfgott (2013)
Harald Helfgott closed the ternary problem completely in 2013, in a pair of long preprints. The strategy was a pincer: push the analytic (circle-method) bounds down to about 1027, and verify all odd numbers below that by computation (the lower verification building on work with David Platt). The two ranges met.
Helfgott (2013): every odd integer n > 5 is a sum of three primes: unconditionally, with no exceptions.The weak / ternary Goldbach conjecture, fully proven. Widely accepted; the full manuscript's formal journal publication has been protracted, but the result is regarded as established.
Direct computation (verification, not proof)
The strong conjecture has been checked by exhaustive computation. Tomás Oliveira e Silva, with Siegfried Herzog and Silvio Pardi, verified it for every even number up to 4 × 1018 (reported 2013). Every single even number in that colossal range is a sum of two primes. This is overwhelming evidence and zero proof. A conjecture about all of ℕ is untouched by any finite check, however vast.
5 · Why it breaks down: the honest crux
Wall 1: The parity problem in sieve theory (why Chen's "1+2" cannot become "1+1"). Sieve methods, which gave Brun's 9+9 and Chen's 1+2, suffer from a structural blind spot identified by Atle Selberg and made precise by Enrico Bombieri: the parity obstruction. A pure sieve cannot distinguish numbers with an even number of prime factors from those with an odd number. It can trap the summand inside the set {primes} ∪ {products of two primes}: because that set is "parity-balanced": but it cannot, on its own, break the tie and force the summand to have exactly one prime factor. Driving Chen's second summand from "at most two prime factors" to "exactly one" (i.e. prime) is exactly the forbidden parity distinction. This is not a matter of working harder or sieving more cleverly; Selberg exhibited model sequences showing the sieve provably cannot close this last step unaided. To cross it you need genuinely non-sieve input: an injection of information about the primes' additive structure that sieves do not see. No such input of the required strength is known.
Wall 2: Minor arcs and the loss of a whole log (why the circle method stops at three primes). In the integral r(n) = ∫ S(α)² e(−nα) dα, the frequencies α split into major arcs (near rationals with small denominator, where S(α) is large and well-understood: these produce the Hardy–Littlewood main term) and minor arcs (everywhere else, which must be shown to contribute a negligible error). For the ternary problem the integrand is S(α)³, and one can bound the minor-arc contribution using ∫₀¹ |S(α)|² dα ≈ N/log N (a known mean value) together with a pointwise bound on one factor of S(α). The arithmetic works: three copies give enough room. For the binary problem the integrand is only S(α)². There is exactly one fewer factor of S to spend, and the minor-arc estimate falls short by a factor of about log N. That missing logarithm is the whole game. A pointwise minor-arc bound strong enough to beat it would essentially require square-root cancellation in S(α) uniformly over all irrational α: which is far beyond what the Riemann Hypothesis itself, let alone current technology, delivers. Three primes have slack; two primes have none.
Wall 3: Average vs. pointwise (the Montgomery–Vaughan ceiling). Every unconditional strong-Goldbach result is a statement about almost all even numbers: the exceptional set E(x) ≪ x1−δ is tiny but not empty-by-proof. The reason is that the circle method, even on GRH, controls the integral on average over n, not for a prescribed individual n. The error term in the asymptotic for r(n) is, for a single n, potentially as large as the main term: so the method cannot guarantee the main term wins at that specific target, only that it wins for the overwhelming majority of targets. Converting an average statement into a pointwise one ("for this even number") is the recurring hard core of analytic number theory, and Goldbach is where it bites hardest: the conjecture is a demand for a pointwise conclusion that our average machinery is structurally unable to supply.
Wall 4: Why GRH is not enough. One might hope the Generalized Riemann Hypothesis: the strongest plausible statement about where the zeros of Dirichlet L-functions lie: would finish the job. It does not. GRH gives essentially optimal control of S(α) on the major arcs, and that is what let Deshouillers–Effinger–te Riele–Zinoviev (1997) prove weak Goldbach conditionally. But GRH says almost nothing new about the minor arcs for the binary problem: the minor-arc deficit of Wall 2 is about equidistribution of primes against irrational frequencies, a question orthogonal to the location of L-function zeros. So even granting the deepest conjecture in the field, strong Goldbach does not fall out. This is a sharp indicator that the problem needs a genuinely new idea, not merely a resolution of the Riemann Hypothesis.
Wall 5: Fringe / speculative routes, and why none has closed it (labelled speculation). The following are honestly-labelled speculative directions, not accepted proofs.
- Elementary / combinatorial "constructions." The internet and journals' slush piles overflow with purported elementary proofs: pairing arguments, density arguments, pigeonhole on residue classes. Every one examined has failed at the same hidden point: it tacitly assumes the primes are equidistributed in short intervals or arithmetic progressions in a way that is itself unknown (often equivalent to statements as hard as or harder than Goldbach). A correct elementary proof is not ruled out in principle, but any that evades Walls 1–3 would itself be a revolution in how we understand primes additively. None has survived scrutiny.
- Probabilistic / "Cramér-model" heuristics. Modeling primes as a random set of density 1/log n makes Goldbach not just true but true with overwhelming margin (Section 3). This is why everyone believes the conjecture. But a heuristic model is not the primes: the Cramér model is known to mispredict some fine statistics, and "a random set of this density would work" is no theorem about the one deterministic set that actually matters. The speculation here is that a sufficiently refined pseudorandomness theorem (in the spirit of Green–Tao's transference principle, which proved arbitrarily long arithmetic progressions in the primes) might one day supply the missing pointwise input. To date, transference-type methods have conquered linear patterns and the ternary (three-variable) problem, but the binary two-variable problem lies just outside their current reach: again for the one-fewer-degree-of-freedom reason of Wall 2.
- Physical / spectral analogies. Occasional proposals recast Goldbach via quantum-chaos or statistical-mechanics analogies (treating r(n) like a partition function). These produce suggestive pictures and match the Goldbach-comet data, but none has yielded a rigorous lower bound r(n) ≥ 1 for all even n: they remain descriptions of the expected behavior, not proofs that no exception exists.
In every case the speculative route either smuggles in an unproven equidistribution assumption, or reproduces the exact minor-arc / parity deficit that stops the classical methods. That consistency across wildly different frameworks is itself evidence that the obstruction is real and structural, not an artifact of one technique.
6 · Status: confirmed
Strong (binary) Goldbach (every even n > 2 is a sum of two primes) is OPEN. It is verified by computation for all even numbers up to 4 × 1018 (Oliveira e Silva, Herzog, Pardi); proven for almost all even numbers with a power-saving exceptional-set bound (Montgomery–Vaughan, 1975); and approached to within "1 + 2" by Chen's theorem (1973). None of these is a proof of the full statement, and the obstructions above explain precisely why.
Weak (ternary) Goldbach (every odd n > 5 is a sum of three primes) is PROVEN (Helfgott, 2013, building on Vinogradov 1937 and large-scale computation). This is a genuine, celebrated resolution: but of the easier of the two problems, and it does not imply the strong form.
What a genuine proof of the strong form would still require
- A way to make the circle method deliver a pointwise lower bound r(n) ≥ 1 for every even n, not merely for almost all n: i.e. to defeat Wall 3.
- Either a minor-arc estimate for S(α)² that recovers the missing factor of log N (Wall 2), or an entirely non-circle-method source of additive information about the primes.
- If pursued by sieves, a parity-breaking input that lets one descend from "at most two prime factors" to "exactly one": something sieve theory provably cannot do by itself (Wall 1).
In short: strong Goldbach is not waiting on a clever rearrangement of known tools. It is waiting on a new way to turn the statistical regularity of the primes into a guarantee about every individual even number. That is the real unsolved problem, and it remains unsolved.
Credit where it is due. Christian Goldbach and Leonhard Euler (1742) for the question and its sharp form; G. H. Hardy and J. E. Littlewood (1923) for the circle method and the asymptotic conjecture; Viggo Brun (c. 1915–1920) for sieve theory and the first 9+9 result; Ivan Vinogradov (1937) for unconditional ternary Goldbach for large odd numbers; Lev Schnirelmann (c. 1930) and Olivier Ramaré (1995) for bounded-sum results; Chen Jingrun (1966/1973) for "1+2," the closest approach to the binary problem; Atle Selberg and Enrico Bombieri for articulating the parity obstruction; Hugh Montgomery and Robert Vaughan (1975) for the exceptional-set bound; Deshouillers, Effinger, te Riele and Zinoviev (1997) for the conditional ternary result; Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi for verification to 4 × 1018; and Harald Helfgott, with David Platt, for proving the weak conjecture in full (2013). The strong (binary) conjecture remains unsolved. It is claimed by no one here: if it is ever solved, that proof is offered freely to the world.
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