argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Prime tuples heuristic

forThis argument, if it holds, bears in favour of the claim.constitution →For every positive even number n, there are infinitely many pairs of consecutive primes differing by n.

The argument

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.

Does it work?

Not yet evaluated.

Premises

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 →Every admissible tuple of integers takes simultaneously prime values infinitely often.

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
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.