Proven partial results
The argument
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.
Does it work?
Not yet evaluated.
Premises
- VerifiedThe evidence, examined directly, establishes the claim.constitution →There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.
- 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 →The even numbers that occur infinitely often as gaps between consecutive primes have positive lower density.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 16, 2026
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.