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The Frontier · ~280 years of attempts

Every branch has been tried. They all hit the same wall.

An honest map of how mathematics has attacked Goldbach — what each approach genuinely proved, and the exact spot it stops. The neighbors have all fallen; the binary case is the lone holdout, for one identified, proven reason.

01Analytic number theory — the circle method

Write the number of representations as an integral and split it into “major arcs” (clean main term) and “minor arcs” (noise). It works beautifully — for three primes.

what it provedTernary Goldbach is DONE: Vinogradov (1937) for large odd numbers, and Helfgott (2013) finished every odd number ≥ 7 as a sum of three primes.
where it stopsWith two variables the minor-arc error terms are as large as the main term. The method needs one more variable than the binary problem gives it.

02Sieve theory — Brun, Selberg, Chen

Cross out composites, count the survivors. This is the branch our whole site visualizes — and it is exactly the one with a proven barrier.

what it provedChen's theorem (1973): every large even number is prime + (prime or a product of two primes). One millimeter away.
where it stopsSelberg's parity problem — a PROVEN theorem that sieves cannot tell apart numbers with an even vs. odd number of prime factors. So they can never force 'exactly one prime.' A ceiling, not a gap in cleverness.

03Additive combinatorics — Schnirelmann, Green–Tao

Study primes as an additive set. You can get “a bounded number of primes” — just not down to two.

what it provedEvery even number is a sum of at most 4 primes (via Helfgott). Green–Tao: the primes contain arbitrarily long arithmetic progressions.
where it stopsBounded-count and AP-structure results don't isolate the binary sum; the pair problem meets the same parity obstruction.

04L-functions and the Riemann Hypothesis

The single most striking honest fact on this page: the biggest hammer in analytic number theory swings and misses on the binary case.

what it provedUnder GRH, ternary Goldbach and strong 'almost all' bounds follow easily.
where it stopsEven assuming the Generalized Riemann Hypothesis — the deepest conjecture in the field — binary Goldbach still does NOT follow. The strongest available tool isn't enough.

05Probabilistic number theory — the Cramér model

Model each number as prime with probability 1/ln. The prediction is enormous and matches the comet — but belief isn't proof.

what it provedPredicts about E / (ln E)² prime pairs for each even E — overwhelmingly positive. This is why everyone believes Goldbach.
where it stopsPrimes aren't actually random, and a heuristic of randomness is not a proof.

06Almost-all results — Montgomery–Vaughan

Statistically we are essentially there. The last step is the hard one: from almost-all to all.

what it provedThe set of even numbers that could fail Goldbach has density zero — exceptions, if any, are vanishingly rare (bounded by X^(1−δ)).
where it stopsUpgrading 'all but a vanishing fraction' to 'every single one' is the entire remaining difficulty.

07Ergodic theory, geometry, spectral methods

The branches that revolutionized nearby problems have no foothold on the pair sum itself.

what it provedErgodic methods cracked Szemerédi and powered Green–Tao. Random-matrix models predict prime correlations with uncanny accuracy.
where it stopsNo known translation of the BINARY sum into a mixing/recurrence or geometric statement; the physics models support but don't prove.

08Logic and computation

If Goldbach is false, it is provably false. If it is true, we may simply need a genuinely new idea — and finitely checking forever won't supply one.

what it provedGoldbach is a Π₁ statement — a counterexample would be a single finite check. Verified by computer to 4×10¹⁸ with no exception.
where it stopsNot known to be independent of ZFC (and no evidence it is). Verification, however vast, can never cover infinitely many numbers.

◆The one wall

Every branch converges on the same obstruction, and it has a name: the parity problem for the two-variable case. The three-variable version is solved — the extra variable gives the circle method the room it needs. So the record reads:

ternary Goldbach (3 primes) .......... PROVED (Helfgott 2013) every even = at most 4 primes ........ PROVED Chen: even = p + (p or p·p) ......... PROVED (1973) almost every even = p + q ............ PROVED (density-zero exceptions) verified to 4×10^18 .................. DONE (by computer) — every even = p + q (binary) ......... OPEN
the honest bottom lineThe binary case isn't open because nobody tried the right branch. It's open because every branch runs into a proven barrier in the same place: distinguishing “exactly one prime” from “a prime times something,” and pinning a prime and its reflection E − p to both be prime at once. That correlation is what no current method can compel.

Which is exactly where this whole site's block/sieve construction arrives — a correct, self-checking picture that runs right up to the frontier and stops honestly at it. That's not failure; that's standing in the right place. Back to the construction →

▪What the data shows

Findings from the experiments run in this project — every number below is from an actual computation over the millions, not a citation. They all point at the same wall from different sides.

The correlation is real — and it's a mod-3 story

If “x is prime” and “E − x is prime” were independent, each even E would have about (E/2)/(ln E)² pairs. The true count is 2× to 4× higher — and the split is clean:

E divisible by 3 → boost ≈ 4× (twice as many pairs) E not divisible by 3 → boost ≈ 2×

The same mod-3 signal showed up independently in the descent search: multiples of 3 solve in one step ~95% of the time; the failed subtractions are eaten by the prime 3 80.9% of the time. Two different experiments, one structure — so it's real.

The comet's floor rises — the margin is widening

Comparing the average number of pairs to the worst case in each window:

near 10,000 avg 202 min 92 min/avg 0.46 near 100,000 avg 950 min 570 min/avg 0.60 near 1,000,000 avg 6163 min 3963 min/avg 0.64

The worst-case even never approaches zero — it holds a robust, rising fraction of the average. The absolute floor (fewest pairs ever) is 1, at E = 6; after that it climbs without bound.

the encouraging halfEvery measurement says Goldbach holds with a margin that grows: pairs are 2–4× more common than chance, and the minimum stays well above zero and rising. Verified with no exception to 4×10¹⁸.
exactly where it stops being provableThe average number of pairs is smooth and provable. The minimum — that it's never zero — is what Goldbach needs, and the average provably does not control the minimum. That single gap between “average is fine” and “every one is fine” is the whole distance from almost all (proved) to all (open). No deterministic rule names the pair either — “largest prime ≤ E/2” fails 78% of the time; you must always search.