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Polignac's conjecture: for every even natural number n, there are infinitely many pairs of consecutive primes differing by exactly n; its case n = 2 is the twin prime conjecture. Claims about the conjecture in general, about a specific fixed even gap occurring infinitely often, and related fixed-gap results belong here; bounded-gap results that fix no particular gap stay under Twin prime conjectur

For every positive even number n, there are infinitely many pairs of consecutive primes differing by n.

SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The even numbers that occur infinitely often as gaps between consecutive primes have positive lower density.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Number theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

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