Prime gaps beyond the sieve parity limit: H₁ < 6 under generalized Elliott–Halberstam
Needs the first parity-breaking argument for prime tuples, which seventy years of sieve theory have not produced, though the target is conditional on GEH and asks only for gap 4. So 80: below the twin prime conjecture (87 in calibration), which needs the same idea without the GEH assumption, and just below rank-two BSD (82); above diagonal Ramsey (78). Application value 2.5 matches twin primes. Leverage 35 is a little above twin primes (29) because a parity-breaking method would be reused across sieve problems.
The problem
Prove that primes come in close pairs under a standard assumption, getting past a known blind spot of sieve methods.
- 1Do primes keep coming in close pairs?
- 2Sieve methods prove a bounded gap
- 3Best possible from sieves alone is 6
- 4Parity blind spot blocks twin primes
Setting. Let be the smallest gap that occurs infinitely often between consecutive primes. The twin prime conjecture is . A set is if it does not cover all residue classes modulo any prime; the prime -tuples conjecture says that every admissible set has infinitely many translates that are simultaneously prime.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Harmonic analysisTheory
- Step 2
- Step 3
- The Riemann zeta function and prime distributionTheory · host
- Then
- Prime gaps beyond the sieve parity limit: H₁ < 6 under generalized Elliott–HalberstamThe problem
State of the art
Open. Chen (1973) proved infinitely many primes p with p + 2 a prime or a product of two primes. Zhang (2013) proved H₁ < 7·10^7; Maynard (arXiv:1311.4600) and Tao independently introduced multidimensional Selberg weights (H₁ ≤ 600, and H₁ ≤ 12 under Elliott–Halberstam); Polymath8b (arXiv:1407.4897) reached H₁ ≤ 246 unconditionally and H₁ ≤ 6 under GEH, and proved that sieve methods cannot go below 6 even under GEH. The unconditional bound moved in late-summer 2026 preprints that are not refereed: Stadlmann (arXiv:2608.31126) H₁ ≤ 240 via equidistribution for smooth moduli; an Axiom Math group claims 212; and an OpenAI manuscript released 3 September 2026 claims 186, with a Lean formalization conditional on stated exponential-sum and numerical assumptions. None of these touches the parity limit. Parity has been broken only for special sequences: Friedlander–Iwaniec (1998) for primes of the form x² + y⁴ and Heath-Brown (2001) for x³ + 2y³, using bilinear (Type II) information. Logarithmically averaged two-point Chowla (Tao, 2016) handles a parity-sensitive correlation but not with the prime weights needed here.
Realistic applications
None for engineering. A parity-breaking mechanism for prime tuples would transform sieve theory and the study of additive patterns in primes (Goldbach-type and twin-type problems); the sieve and bilinear technology is used throughout analytic number theory.
What counts as solved
A refereed proof that H₁ ≤ 4 under GEH (or under Elliott–Halberstam, or unconditionally), or of the twin prime conjecture under any standard distributional hypothesis. Improvements of the unconditional bound 246 do not count, since they stay within the sieve framework.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The target is H_1 <= 4 under generalized Elliott-Halberstam, or any route past the sieve parity limit. Formal Conjectures has Elliott-Halberstam and twin-prime statements (Wikipedia/ElliottHalberstamConjecture.lean) and a prime-gap helper file, but a new method beyond the sieve has no formal template. Experts referee it; improvements to the unconditional 246 do not count.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
A new idea for detecting parity is needed; computation helped optimize sieve weights for the existing bounds but cannot cross the parity limit.
Bottlenecks
ranked by votes- #1TheoryParity barrier
Selberg’s example, adapted by Polymath8b, shows that sieve weights with level-of-distribution input, even under GEH, cannot distinguish integers with an even or odd number of prime factors, so cannot force two primes among h₁, h₂ apart by less than 6.
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Connected problems
- This problem is related to Logarithmically averaged Chowla conjecture for even k ≥ 4: Σ λ(n)λ(n+1)λ(n+2)λ(n+3)/n = o(log x)
Progress
- Aug 31, 2026Unconditional bound 240 (preprint)source
Stadlmann; AI-produced claims of 212 and 186 followed, all unrefereed and all within the sieve framework.
- Jul 18, 2014Polymath8b: 246, 6 under GEH, and the parity limitsource
Shows that 6 is the best a sieve argument can give under GEH.
- Nov 19, 2013Maynard’s multidimensional sievesource
H₁ ≤ 600, and H₁ ≤ 12 under Elliott–Halberstam.
- Jan 1, 1973Chen’s theoremsource
p + 2 is a prime or a product of two primes infinitely often (date approximate to the year).