There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is Yitang Zhang's 2013 theorem on bounded gaps between primes, the first proof that some fixed difference between primes recurs infinitely often. Zhang's paper, refereed and published in the Annals of Mathematics in 2014, proves that the smallest gap between consecutive primes falls below 70 million infinitely often; since only finitely many differences lie below that bound, at least one of them must be the difference of infinitely many prime pairs. The proof rests on two ingredients: a Goldston–Pintz–Yıldırım-type sieve, which Zhang showed can be restricted to moduli free of large prime factors, and a new estimate showing that primes are well distributed in arithmetic progressions to such smooth moduli slightly beyond the classical one-half barrier, since any distribution level beyond one half yields bounded gaps.
The result is accepted without dissent and has been independently confirmed by a stronger route. Within months, James Maynard and Terence Tao developed a different multidimensional sieve that needs only the classical Bombieri–Vinogradov theorem and gives a bound of 600, and the Polymath8b collaboration refined it to show that infinitely many pairs of primes differ by at most 246. Any of these bounds implies the claim by the same pigeonhole step. Machine-checked formalizations of the 246 bound conditional on Bombieri–Vinogradov were released in 2026, and unconditional versions are in progress, but the verdict here rests on the refereed proofs, not on those checks.
The specific constant 70 million has no special significance; Zhang chose it for convenience, and the claim is a weak corollary of what is now known. The theorem is a proven special case of the still-open conjecture that infinitely many pairs of primes differ by exactly 2, which it does not settle.
Full reasoning: the evidence and decisions behind this verdict
The claim is the pigeonhole corollary of Zhang's main theorem, stated in his abstract as the assertion that the lower limit of consecutive prime gaps is below 7 × 10^7 (Y. Zhang, "Bounded gaps between primes", Annals of Mathematics 179 (2014), 1121–1174; the abstract is recorded as an instance at archive.org/details/yitangzhangpaper). If consecutive primes differ by less than 70 million infinitely often, then since there are only finitely many positive integers below 70 million, some one of them is the difference of infinitely many prime pairs; conversely the claim as stated, about pairs of primes rather than consecutive ones, follows at once because a pair (p, p + N) both prime forces the consecutive gap after p to be at most N. The two formulations are equivalent in content.
Two independent lines establish it. First, Zhang's own proof: a Goldston–Pintz–Yıldırım-type sieve restricted to smooth moduli, together with a Bombieri–Vinogradov-type estimate at level 1/2 + 1/584 for smooth moduli with well-factorable weights, proved by the dispersion method and Deligne-type bounds on exponential sums. The paper was refereed by the Annals unusually quickly, its argument was re-derived and expounded in detail by the Polymath8a project ("New equidistribution estimates of primes in arithmetic progressions", Research in the Mathematical Sciences 1 (2014)), which improved the constant to 4680, and by expository surveys by Granville (Bulletin of the AMS 2015) and Green (arXiv:1402.4849). No objection to its correctness has ever been raised. The material subclaims, that primes have level of distribution beyond one half to smooth moduli for well-factorable weights and that any distribution level beyond one half yields bounded gaps, are both established theorems (the latter is Goldston, Pintz and Yıldırım's 2009 Annals result, with Zhang's adaptation to smooth moduli).
Second, the Maynard–Tao sieve: Maynard ("Small gaps between primes", Annals of Mathematics 181 (2015), 383–413) and independently Tao developed a multidimensional Selberg sieve that, using only the classical Bombieri–Vinogradov theorem, gives infinitely many consecutive primes differing by at most 600; Polymath8b ("Variants of the Selberg sieve, and bounded intervals containing many primes", Research in the Mathematical Sciences 1 (2014), arXiv:1407.4897) lowered this to 246. That infinitely many pairs of primes differ by at most 246 implies the claim by the same pigeonhole step and does not use Zhang's distribution theorem, so the two lines corroborate each other independently. The Polymath8b abstract itself states Zhang's bound as an established result and is recorded as an instance.
Machine-checked evidence is partial and not relied on. Axiom Math released a Lean 4 formalization (github.com/AxiomMath/PrimeGapsLib) proving the 246 and 600 bounds conditional on the Bombieri–Vinogradov theorem, taken as a hypothesis; a pull request integrating a Bombieri–Vinogradov formalization to make it unconditional was open and unreviewed at the time of this pass, and a separate FormalPantheon project reports a formalization based on Maynard and Zhang with Bombieri–Vinogradov as an intermediate theorem. None of these has been checked by the graph, and none establishes the claim unconditionally on its own; they are consistent with the informal record and add modest corroboration.
Instances: all three recorded sources (Zhang's abstract, the Polymath8b abstract, and the Quanta feature reporting the result) affirm the claim; no source denies it, and the discourse contains no dissent. The verdict is "verified" in the accepted-proof sense of the mathematics standard: a refereed proof, independently expounded, standing without objection, and confirmed by independent stronger results. What would change the conclusion: an error found in both Zhang's argument and the Maynard–Tao argument, which after twelve years of scrutiny, expository re-derivation, and partial formalization is remote. The residual credence gap reflects only the generic possibility of undetected error in a long analytic proof.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
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.
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.
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.
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.
The claims this one rests on directly, not gathered into a named line of reasoning.
- specifiesa more specific version of the parentsteward instructions →There are infinitely many pairs of primes that differ by exactly 2. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N.
Describing Zhang's result on bounded gaps between primes.
The source's own evidence bears what it asserts. The Quanta feature states Zhang's theorem accurately and rests it on the paper itself and on the referee report and named number theorists (Granville, Goldston) it quotes; it offers no evidence of its own beyond that reportage, and its description of the proof's structure (a GPY-type sieve restricted to moduli free of large prime factors) matches the paper.
It is proved that lim inf n→∞ (p_{n+1} − p_n) < 7×10^7, where p_n is the n-th prime.
Abstract of Zhang's paper (Annals of Mathematics 179 (2014), 1121–1174), stating the main theorem, which by pigeonhole is the proposition that some N below 70 million is the difference of infinitely many prime pairs.
A celebrated recent result of Zhang showed the finiteness of H_1, with the explicit bound H_1 ≤ 70000000.
Abstract of the Polymath8b paper, recounting Zhang's bound as an established result before improving it to 246.
How these sources relate
- https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/ draws its statement from Bounded gaps between primes, faithfully. The article reports the content of Zhang's Annals paper; its statement of the result is the pigeonhole consequence of the paper's theorem (liminf of consecutive prime gaps below 7×10^7) and neither strengthens nor hedges it. Only the paper's abstract was opened here, not the full text. Judged from the citing document alone.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.