Zhang's proof (2013)
The argument
Zhang's Annals paper establishes that primes have a level of distribution beyond one half in arithmetic progressions to smooth moduli, for well-factorable weights, a Bombieri–Vinogradov-type estimate at level 1/2 + 1/584 obtained by dispersion and Deligne-type exponential-sum bounds. Because any level of distribution exceeding one half yields infinitely many bounded gaps between primes by the Goldston–Pintz–Yıldırım sieve, and because Zhang showed the sieve can be restricted to smooth moduli without loss so that his weaker distribution estimate suffices, the two combine with an explicit admissible tuple of k = 3.5 × 10^6 primes to give infinitely many consecutive primes differing by less than 7 × 10^7; by pigeonhole some single difference below 70 million then recurs infinitely often.
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 →Any level of distribution for primes in arithmetic progressions exceeding 1/2 implies infinitely many bounded gaps between primes.
- 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 →Primes have a level of distribution exceeding 1/2 in arithmetic progressions to smooth moduli, for well-factorable weights.
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.
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Earlier ratings
- Sep 16, 2026
The inference goes through: given the two ingredients, the restricted GPY sieve and Zhang's adaptation of it to smooth moduli yield the explicit bound and the pigeonhole step is immediate. The argument rests on Zhang's distribution theorem for smooth moduli beyond one half, the paper's genuinely new contribution, and on the Goldston–Pintz–Yıldırım criterion that any level beyond one half gives bounded gaps, which was already a theorem in 2009; both are refereed results that have stood without objection and were re-derived in full by the Polymath8a project.
- Sep 16, 2026
The inference is valid: a GPY-type sieve with a level of distribution beyond one half for smooth moduli produces an admissible tuple of width below 70 million containing two primes infinitely often, and pigeonhole gives a single difference N. The argument stands on Zhang's distribution theorem for smooth moduli, the genuinely new step, which was refereed in the Annals and independently re-derived and sharpened by the Polymath8a project; the Goldston–Pintz–Yıldırım conditional theorem it relies on is a settled 2009 result. Neither premise faces any objection in the literature.