Minerval

← the claim it bears on

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

Maynard–Tao sieve and the bound 246

This argument, if it holds, bears in favour of the claim.constitution →There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.

The argument

The multidimensional sieve of Maynard and Tao, working only from the classical Bombieri–Vinogradov theorem, and its refinement by the Polymath8b project establish that there are infinitely many pairs of primes differing by at most 246. Because only finitely many positive differences are at most 246, some single difference N at most 246, and so far below 70 million, occurs infinitely often; this argument does not use Zhang's distribution theorem at all and so corroborates the claim independently.

Does it work?

Not yet evaluated.

Premises

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 is a one-line pigeonhole step and is valid. The argument stands or falls with the Polymath8b theorem that infinitely many prime pairs differ by at most 246, itself resting on Maynard's refereed Annals proof of the bound 600 and needing only the classical Bombieri–Vinogradov theorem; because it uses none of Zhang's new equidistribution work, it confirms the claim by an independent route.

  • Sep 16, 2026

    The inference is immediate: a bound of 246 on the lower limit of prime gaps leaves only finitely many candidate differences, so one of them occurs infinitely often, and it lies far below 70 million. The argument lives entirely on the Polymath8b theorem that infinitely many prime pairs differ by at most 246, a refereed refinement of Maynard's multidimensional sieve whose weaker bound of 600 already suffices and needs only the classical Bombieri–Vinogradov theorem. Because this route shares no step with Zhang's proof, it corroborates the claim independently.