For every positive even number n, there are infinitely many pairs of consecutive primes differing by n.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim is an open conjecture of analytic number theory, so the assessment rests on the state of proof and partial results rather than on the instances, of which the graph holds one: a 2013 Quanta Magazine feature recounting Polignac's conjecture as background to Zhang's theorem (www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/). That source paraphrases the conjecture loosely (omitting "even" and "consecutive") and takes no position of its own; it carries no evidential weight beyond confirming how the conjecture is attributed.
Status of the proposition. Standard references (for example en.wikipedia.org/wiki/Polignac's_conjecture) state it in the consecutive-gap form adopted here and note that it has not been proven or disproven for any given n. The two lines of support weigh as follows. First, the proven partial results: Zhang's bounded-gaps theorem (Annals of Mathematics, 2014), improved to 246 by Maynard and Polymath8b, gives infinitely many prime pairs with difference at most 246, and since the consecutive gaps inside such pairs are bounded, at least one even number at most 246 recurs infinitely often as a consecutive-prime gap. Pintz's positive-density theorem (2013, building on Zhang's level-of-distribution estimate and the Goldston–Pintz–Yıldırım sieve) shows the set of such gaps has positive lower density, and Granville and collaborators derived explicit lower bounds on the proportion of even integers that are differences of infinitely many prime pairs from the Maynard–Tao sieve (arxiv.org/abs/1410.8198). These establish the conjecture for a positive proportion of even numbers in the aggregate while identifying none individually, which is exactly the shape one expects from sieve methods blocked by the parity problem. Second, the heuristic line: the Hardy–Littlewood prime tuples conjecture implies the consecutive form by choosing admissible tuples that force the intermediate integers to be composite, and its quantitative version predicts counts of prime pairs with each gap that match computation to high precision. This is conditional support, since the tuples conjecture is itself open, but it is the accepted organising heuristic of the field.
Against: nothing. No counterexample is conceivable in finite computation (the claim is about infinitude), no heuristic model predicts failure for any gap, and no credible mathematician asserts the negation. Several self-published "proofs" of Polignac's conjecture circulate (for instance on ResearchGate); none has been accepted by the literature and they do not change the status.
Verdict. Under the mathematics conventions, an open conjecture with proven weaker statements and heuristic evidence mathematicians count is supported rather than unsupported, and it is not contested because no credible party disputes it. Credence 0.95 reflects near-universal expert belief, the Hardy–Littlewood agreement, and the partial theorems, tempered by the fact that no single gap is settled and that the twin prime case, the weakest instance, has resisted proof for over a century; the credence here must not exceed that of the twin prime conjecture, since this claim implies it. A proof for any specific even n, or a proof of the tuples conjecture, would raise the status; a machine-checked proof of the recorded formal statement would verify it. Refutation would require exhibiting an even n with only finitely many consecutive-prime gaps of that size, which no known method approaches.
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.
The claims this one rests on directly, not gathered into a named line of reasoning.
- specifiesa more specific version of the parentsteward instructions →There are infinitely many pairs of primes that differ by exactly 2. ↗︎
Because infinitely many prime pairs differ by some fixed N below 70 million (a bound since lowered to 246), at least one even number is a gap between consecutive primes infinitely often, and because the even numbers recurring infinitely often as consecutive-prime gaps have positive lower density, the conjecture is known to hold for a positive proportion of even numbers; these proven weaker statements raise confidence that it holds for every even number, without establishing it for any particular one.
The inference from the premises to their stated conclusions is sound: the bounded-gaps theorem does yield at least one even number recurring infinitely often as a consecutive-prime gap, and Pintz's positive-density theorem does extend this to a positive proportion of even numbers. Both premises are accepted theorems of the literature, so the argument's weight is secure. The caveat is in reach rather than validity: these results support the conjecture for every even number only by analogy and momentum, since they identify no particular gap and sieve methods of this kind are known to stop short of any specific one.
Given that every admissible tuple of integers is simultaneously prime infinitely often, for each even n one may take an admissible tuple containing 0 and n whose remaining entries force every integer strictly between p and p + n to be composite, so infinitely many simultaneous prime values yield infinitely many consecutive primes with gap exactly n; the conjecture therefore follows from the prime tuples conjecture, whose quantitative form also predicts the observed counts of prime pairs with each gap.
Granting the premise, the inference goes through: for each even n an admissible tuple containing 0 and n can be chosen so that the remaining entries force every integer strictly between the two primes to be composite, so the conjecture is a consequence of the prime tuples conjecture. The argument therefore lives or dies entirely on that premise, which is itself an open conjecture, so what it transfers to Polignac's conjecture is the tuples conjecture's heuristic and numerical support rather than a proof.
Provenance
Where this claim has been said, linked to its canonical form.
In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.
Generalization of the twin primes conjecture.
Asserted without evidence of the source's own. A news feature on Zhang's 2013 bounded-gaps theorem. It recounts Polignac's 1849 conjecture in loose paraphrase, without the restriction to even gaps or to consecutive primes, and offers no evidence for it; the article's own subject is the proven weaker result that some gap below 70 million recurs infinitely often.
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.