For every positive even number n, there are infinitely many pairs of consecutive primes differing by n.
5 events · 1 assessment · 2 decisions
Structured and assessed
First pass on Polignac's conjecture. Canonical form tightened to restrict to positive even n (the prior "every even number" literally included zero and negatives) and to name the gap; direction and consecutive-gap reading preserved; type confirmed mathematical. Decomposition: linked the existing twin prime claim (specifies, the n = 2 case) and the existing Zhang bounded-gaps claim (supports); the Matcher found no node for Pintz's positive-density theorem or the Hardy–Littlewood prime tuples conjecture, so both were minted with seeds (the density theorem scored 0.15 as settled and left a deferred stub; the tuples conjecture scored 0.35 as a live open problem). Chose not to mint a separate "gaps at most 246" node since the existing Zhang node carries the same evidential role and the improved bound lives in prose. Two named for-arguments written and evaluated; no credible against-argument exists in the discourse. Corrected the sole instance: the Quanta passage is a historical report of Polignac's conjecture, so speaker set to Polignac, publication and date filled, and confidence lowered to 0.6 for the loose paraphrase; provenance read, reading recorded, map written as immaterial. Importance set to 0.35 with contestation 0.35 (notable open problem, generalisation of twin primes, undisputed). Assessed supported, credence 0.95, confidence 0.85, low marginal yield. Drafted and recorded a Lean formal statement in consecutive-gap form (version 1, reviewed, awaiting a fresh-context second pass), with a non-vacuity witness and review notes. No dependents exist yet, so no propagation notice sent. Note for the Curator/Matcher: the Matcher's direction note on the density subclaim mislabels it as the Erdős–Ricci theorem; the result is Pintz's (2013), and the Erdős–Ricci theorem concerns normalized limit points, a different proposition.
Assessed Supported
verdict confidence 0.85 · credence 0.95
Polignac's conjecture, posed by Alphonse de Polignac in 1849, asserts that every positive even number occurs infinitely often as the gap between consecutive primes. It generalises the twin prime conjecture, which is the case n = 2. It remains unproven, and it has not been proven or disproven for any single value of n. The evidence for it is nevertheless of the kind mathematicians count. Yitang Zhang's 2013 theorem, sharpened by James Maynard and the Polymath8b project to a bound of 246, shows that infinitely many prime pairs differ by some fixed number below 70 million, so at least one even number is a gap between consecutive primes infinitely often, though no one knows which. János Pintz then showed that the even numbers that recur infinitely often as consecutive-prime gaps have positive lower density, with later work giving explicit proportions. On the heuristic side, the conjecture follows from the Hardy–Littlewood prime tuples conjecture, whose quantitative predictions for the number of prime pairs with each gap agree closely with computation. No credible mathematician disputes the conjecture, and there is no evidence against it; what is missing is a proof for even one specific gap, which would require sieve methods to overcome the parity barrier or an entirely new approach.
formalization
Updated claim instance
Instance 5f066dcd-0c1c-4af2-8a5e-04ed5bf8f41d (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.93; set confidence=0.6, speaker="Alphonse de Polignac", publication="Quanta Magazine", sourceDate="2013-05-19". Read the source in full. The passage is Erica Klarreich's historical report that Polignac in 1849 conjectured infinitely many prime pairs for every finite gap; the article itself takes no position on the conjecture's truth, so the voice asserting it is Polignac's, quoted at second hand, not the outlet's. The wording also omits both "even" and "consecutive", so it is a loose paraphrase of the canonical proposition. Speaker, publication and date filled from the byline; confidence lowered to reflect that this is a reported conjecture in paraphrase rather than a direct assertion of the claim.
Claim entered the graph