Every even number greater than two is the sum of two primes.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
This is Goldbach's conjecture, posed in correspondence between Christian Goldbach and Leonhard Euler in 1742 and still unproven. No proof and no counterexample is known, so it remains an open problem; but it is an open problem with an unusually large body of evidence in its favour, and mathematicians almost universally expect it to be true.
The evidence is of three kinds. First, exhaustive computation: every even number up to 4 × 10^18 has been checked and found to be a sum of two primes, so any counterexample would be astronomically large. Second, proven approximations: every odd number greater than five is a sum of three primes (the weak Goldbach conjecture, proved by Helfgott in 2013), every sufficiently large even number is a prime plus a number with at most two prime factors (Chen's theorem), and almost all even numbers are sums of two primes, the exceptions having density zero. Third, the Hardy–Littlewood heuristic, under which the number of ways to write an even number as a sum of two primes grows without bound, so that a large even number with no representation at all would defy a rapidly increasing expected count; the computational data agree closely with this prediction throughout the checked range.
None of this amounts to proof. The circle method that proves the three-prime case loses too much information to handle two primes, and sieve methods stall at Chen's "prime plus almost-prime". What would settle the question is a proof, or a counterexample beyond 4 × 10^18, and neither is in sight. The conjecture is stated here in its modern form, with two allowed as a prime and repeated primes permitted, so that 4 = 2 + 2 counts.
Full reasoning: the evidence and decisions behind this verdict
The claim is the binary (strong) Goldbach conjecture. Status follows the mathematics convention for open problems: a conjecture backed by evidence mathematicians count is "supported", not "unsupported", and this one carries more such evidence than almost any other open problem.
Direct evidence weighed. (1) Computational: Oliveira e Silva, Herzog and Pardi, Mathematics of Computation 83 (2014), verified the conjecture for all even numbers 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 architectures (arxiv.org/pdf/2603.07850) still calls this record "the benchmark", so no larger accepted verification exists as of this pass. This is the subclaim verified by computation up to 4 × 10^18. (2) Proven weakenings: the ternary Goldbach theorem (Vinogradov 1937 for large odd numbers; Helfgott 2013 for all odd numbers above five), Chen's theorem (1966/1973), and the density-zero exceptional set theorem (Estermann, van der Corput, Chudakov, late 1930s; Montgomery and Vaughan 1975 with a power saving). All three are settled textbook mathematics and none has met an obstruction that would suggest the full conjecture fails. (3) Heuristic: the Hardy–Littlewood asymptotic for the two-prime representation count of an even n, of order n divided by the square of its logarithm times a singular factor, implies the representation count tends to infinity; the verification paper reports that the observed counts of minimal Goldbach partitions are in excellent accord with the prime k-tuple predictions. This heuristic is itself conjectural and is weighed as plausibility, not proof.
Against: nothing. No credible mathematician asserts a counterexample, and the numerous amateur "proofs" and "disproofs" in the preprint literature are not counted as evidence in either direction. The instance set gives no independent signal: the one recorded appearance (Quanta Magazine, 2013) is a neutral mention of the conjecture as an unsolved problem, not an assertion, and has been recorded as such.
Credence 0.97: the evidence is exactly the kind that has, historically, been reliable for additive problems of this shape (the analogous three-prime asymptotic is a theorem), and the only route to falsity is a counterexample beyond 4 × 10^18 against a rapidly growing expected representation count. The residual reflects that heuristics of this kind have occasionally failed elsewhere in number theory (Mertens, Pólya) and that the conjecture is not proved. Verdict confidence 0.9: the status is a settled reading of an uncontroversial evidential situation. What would change the verdict: a refereed proof (to verified) or a machine-checked proof of the recorded formal statement; a counterexample (to contradicted).
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.
- supportsthis provides evidence for the parentsteward instructions →Every odd number greater than five is the sum of three primes. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Every sufficiently large even number is the sum of a prime and a number with at most two prime factors. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Almost all even numbers are the sum of two primes, in the sense that the exceptional set has density zero. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Goldbach's conjecture has been verified by computation for all even numbers up to 4 × 10^18. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The number of representations of an even number as a sum of two primes tends to infinity as the number tends to infinity. ↗︎
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.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.