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The size of gaps between consecutive primes: the average spacing near x (≈ log x, e.g. the 'about 2.3 × digit count' rule of thumb), maximal and record gaps, Cramér's conjecture and probabilistic models of gap growth, and first occurrences of a given gap size. Claims that a specific fixed gap occurs infinitely often stay under Polignac's or Twin prime conjecture.

The average gap between primes near a large number is approximately 2.3 times its digit count.

VerifiedThe evidence, examined directly, establishes the claim.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 theory Prime number theoremimportance · settledImportance 0.12, 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 →

There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.

VerifiedThe evidence, examined directly, establishes the claim.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 theory Twin prime conjectureimportance · minorImportance 0.25, 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 →

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