Every even number greater than two is the sum of two primes.
Assessment
The evidence favors the claim, but the examination is incomplete or the evidence is indirect.
This is the binary, or strong, Goldbach conjecture, posed by Christian Goldbach in 1742. It is one of the oldest unsolved problems in number theory. It remains unproven, but it is backed by as much evidence as mathematicians can give short of a proof.
Two independent lines of evidence support it. The first is computational. The conjecture has been verified by computer for every even number up to 4 × 10^18. The Hardy–Littlewood heuristic also predicts that the number of ways to write an even number as a sum of two primes grows without bound, and observed counts match that prediction closely. The second line is theoretical. Proven results come close to the conjecture from several directions: the ternary Goldbach theorem, Chen's theorem on sums of a prime and a number with at most two prime factors, and the theorem that the exceptions have density zero.
No credible mathematician argues that the conjecture is false. Nothing proven implies it either, because the sieve and circle-method techniques behind the partial results run into the parity barrier at the decisive step. The conjecture would be settled by a refereed or machine-checked proof, or by a counterexample, which would have to be larger than 4 × 10^18.
Full reasoning: the evidence and decisions behind this verdict
Status follows the mathematics convention for open problems. A conjecture backed by evidence that mathematicians count is rated "supported", not "unsupported".
Computational evidence: Oliveira e Silva, Herzog and Pardi, Mathematics of Computation 83 (2014), verified the conjecture up to 4 × 10^18, with an independent double-check to 4 × 10^17 (sweet.ua.pt/tos/bib/4.12.html). A 2026 arXiv preprint on GPU verification (arxiv.org/pdf/2603.07850) still treats this record as the benchmark.
Heuristic evidence: the Hardy–Littlewood asymptotic, of order n divided by the square of log n times a singular series, implies that the representation count tends to infinity. This asymptotic is itself conjectural, so it is weighed as plausibility, not proof.
Proven weakenings: Vinogradov (1937) and Helfgott (2013) for the ternary case; Chen (1966/1973); Estermann, van der Corput and Chudakov for the density-zero exceptional set, with Montgomery and Vaughan (1975) giving a power saving. These are settled theorems. They corroborate the conjecture but do not imply it, because of the parity barrier.
Both named arguments are evaluated as holding with caveats. One is inductive; the other is corroborative only.
Against the claim: nothing credible. Amateur proofs and disproofs are not counted in either direction. The only recorded instance, from Quanta Magazine in 2013, mentions the conjecture neutrally as an open problem, so it has low confidence as an assertion.
Credence is 0.97. The residual 0.03 reflects that heuristics of this kind have occasionally failed elsewhere in number theory (Mertens, Pólya) and that the claim is unproven. This pass found no change to any premise since the last assessment.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Proven results come close to the conjecture from several directions: every odd number greater than five is a sum of three primes, every sufficiently large even number is a prime plus a number with at most two prime factors, and the even numbers that are not sums of two primes form a set of density zero. Since each weakening has been established without meeting any obstruction pointing toward a counterexample, they raise confidence in the binary conjecture, though none of them implies it.
Because the conjecture has been checked by computer for every even number up to 4 × 10^18, any counterexample would have to lie beyond that bound; and because the Hardy–Littlewood heuristic predicts that the number of two-prime representations of an even number grows without bound, with observed counts matching the prediction, a counterexample at larger sizes would require the representation count to collapse to zero against a rapidly growing expectation, which makes the conjecture highly probable.
Provenance
Where this claim has been said, linked to its canonical form.
the Goldbach conjecture, which proposes that every even number is the sum of two primes
Listing oldest unsolved problems about primes and addition.
Asserted without evidence of the source's own. The article names the conjecture as an unsolved problem and describes what it proposes; it does not claim the conjecture is true and offers no evidence on it. It also notes, correctly, that a weaker version was settled by Helfgott in 2013.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.