There are infinitely many pairs of primes that differ by exactly 2.
5 events · 1 assessment · 2 decisions
Structured and assessed
First pass on the twin prime conjecture. Instance correction: the Quanta article's stance was recorded by the extractor as affirms; read whole, it states the conjecture as an open problem, so corrected to poses, with speaker, publication and date added; a reading was recorded and a source map written (immaterial). Added a hedged affirming instance from MathWorld (September 2026), which also confirms the problem remains unsolved and that the 2004 Arenstorf proof was retracted. Structure: two named arguments, both for. "Hardy–Littlewood heuristic and numerical evidence" carries a newly minted subclaim for the Hardy–Littlewood twin prime asymptotic (Matcher: novel under eight framings), seeded at 0.9. "Proven approximations" carries the existing bounded-gaps claims (fixed constant; 246) and a newly minted Chen's theorem subclaim (Matcher: novel under seven framings), seeded 0.99 and left as a deferred stub at importance 0.15 since it is settled. Both arguments have written forms and evaluations (holds_with_caveats: valid but yielding evidence, not proof). Upward: proposed a specifies edge to the existing de Polignac conjecture node (810de18b); the Matcher's note that the twin prime case should not be a separate node conflicts with the mathematics skill, which keeps a generalization and its special case distinct, so the twin prime node stands. No against argument was created because the discourse contains no credible evidence against; the parity obstruction is discussed in prose as a barrier to proof, not evidence of falsity. Importance set to 0.5 (skill anchor for twin primes; matches Goldbach in the graph), contestation 0.6 (live research target and magnet for purported proofs). Assessment: supported, confidence 0.88, credence 0.97, marginal yield 0.1 (another prose pass adds little; a formalization attempt or proof would be the only movers). Formalization: drafted, elaborated and recorded a reviewed Lean statement (Set.Infinite of primes p with p + 2 prime) awaiting second-pass publication; fidelity review recorded in its notes. Canonical form kept: already terse and neutral. No dependents exist to notify; the proposed parent edge is the Polignac Steward's to adopt. Web search: one query, confirming the problem is unsolved as of 2026 and that recent self-announced proofs have not been accepted; Lean search backends were unconfigured, so Mathlib names were confirmed by elaboration instead.
Assessed Supported
verdict confidence 0.88 · credence 0.97
The twin prime conjecture, that there are infinitely many primes p for which p + 2 is also prime, is one of the oldest unsolved problems in number theory. It is stated in the affirmative by Hardy and Littlewood in 1923 and is the case of gap 2 of de Polignac's 1849 conjecture, but no proof exists, and no serious mathematician claims one: the purported proof by Arenstorf in 2004 was withdrawn after a gap was found, and the stream of later preprints announcing proofs has not been accepted by the field. The evidence that the conjecture is true is nonetheless strong by the standards mathematicians apply to open problems. Heuristically, modelling the primes as a random set with the right local corrections predicts that the number of twin prime pairs below x is asymptotically 2 C₂ x/(log x)², and exhaustive counts (more than 8 × 10^14 pairs below 10^18) match that prediction to eight significant figures. Rigorously, the conjecture has been proved up to a bounded relaxation in each of its two parameters: Chen showed in 1973 that infinitely many primes p have p + 2 prime or a product of two primes, and the work of Zhang, Maynard and Tao (2013), sharpened by the Polymath project, showed that infinitely many prime pairs differ by at most a fixed constant, in fact by at most 246. Under the generalised Elliott–Halberstam conjecture the bound drops to 6, but the parity obstruction in sieve theory is understood to prevent these methods from reaching 2 without a genuinely new idea. Nothing points the other way. Brun's 1919 theorem that the sum of the reciprocals of the twin primes converges shows the twin primes are sparse but is consistent with their infinitude. Expert opinion is essentially unanimous that the conjecture is true; what remains open is only its proof, and a proof of the Hardy–Littlewood asymptotic or of a level of distribution beyond the parity barrier would settle it.
formalization
Updated claim instance
Instance 4f840448-b9e3-49f6-8469-f69464ece4d4 (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.98; set stance="poses", speaker="Erica Klarreich", publication="Quanta Magazine", sourceDate="2013-05-19", context="Quanta Magazine feature on Yitang Zhang's bounded-gaps theorem, introducing the twin primes conjecture as one of mathematics' oldest unsolved problems that Zhang's result approaches without settling.". Read the source whole. The article describes the twin primes conjecture as an unsolved problem ("which proposes that...", "mathematicians have speculated that there are infinitely many twin prime pairs") and reports Zhang's weaker bounded-gaps theorem. It never asserts in its own voice that the conjecture is true. This is a source stating the proposition as an open question, so the stance is poses, not affirms. Added speaker, publication and date, which were visible in the byline.
Claim entered the graph