| { |
| "schema_version": "1.0.0", |
| "id": "problem:twin-prime-conjecture", |
| "record_kind": "problem", |
| "slug": "twin-prime-conjecture", |
| "title": "The twin prime conjecture", |
| "field": { |
| "main": "math", |
| "sub": "number theory", |
| "slug": "number-theory", |
| "name": "Number theory", |
| "original_discipline": "mathematics" |
| }, |
| "additional_fields": [], |
| "msc2020": [ |
| { |
| "code": "11", |
| "basis": "community-map", |
| "evidence": "Inferred from community field: number-theory" |
| } |
| ], |
| "problem_statement": "Prove or disprove: there are infinitely many primes $p$ for which $p+2$ is prime. The quantitative form, due to Hardy and Littlewood, predicts\n$$\\pi_2(x) \\sim 2\\Pi_2 \\int_2^x \\frac{dt}{(\\log t)^2}, \\qquad \\Pi_2 = \\prod_{p>2}\\left(1 - \\frac{1}{(p-1)^2}\\right).$$", |
| "status": "open", |
| "source": { |
| "key": "community", |
| "name": "Community consensus", |
| "type": "problem-list", |
| "url": null, |
| "citation": null, |
| "doi": null, |
| "arxiv_id": null, |
| "license": "Original wording", |
| "notes": "Questions that are widely regarded as open but have no single canonical citation; written here from the surrounding literature." |
| }, |
| "lean4_statement": "import Std\n\n-- Original Frontier Science preview transcription; not an upstream formalization.\n-- A statement only: no proof or claim of verification.\ndef TwinPrimeConjecture : Prop :=\n ∀ N : Nat, ∃ p : Nat,\n N < p ∧\n (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧\n (2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → d = 1 ∨ d = p + 2)", |
| "proof": null, |
| "lean4_proof": null, |
| "proof_status": "not-recorded", |
| "lean4_proof_status": "not-recorded", |
| "lean4_target": "∀ N : Nat, ∃ p : Nat, N < p ∧ (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧ (2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → d = 1 ∨ d = p + 2)", |
| "formalizations": [ |
| { |
| "language": "Lean 4", |
| "code": "import Std\n\n-- Original Frontier Science preview transcription; not an upstream formalization.\n-- A statement only: no proof or claim of verification.\ndef TwinPrimeConjecture : Prop :=\n ∀ N : Nat, ∃ p : Nat,\n N < p ∧\n (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧\n (2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → d = 1 ∨ d = p + 2)", |
| "target": "∀ N : Nat, ∃ p : Nat, N < p ∧ (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧ (2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → d = 1 ∨ d = p + 2)", |
| "source_url": null, |
| "format": "source-file", |
| "sha256": "4e4ec9454666e0f1090795d031a14397289e67b37674bfc6f5c7081c688ffeb7", |
| "license": "Original wording", |
| "toolchain": null, |
| "commit": null, |
| "has_sorry_or_admit": false, |
| "has_axiom_declaration": false, |
| "verification_status": "not-checked", |
| "correspondence_status": "not-checked" |
| } |
| ], |
| "contributions": [], |
| "teaser": "Are there infinitely many primes p such that p + 2 is also prime?", |
| "background": null, |
| "partial_results": "- Zhang (2013) proved that $\\liminf_n (p_{n+1}-p_n) < 7\\times 10^7$, the first finite bound.\n- Maynard and, independently, Tao gave a different sieve; the Polymath8 project reduced the bound to $246$ unconditionally, and to $6$ under the generalised Elliott–Halberstam conjecture.\n- Chen (1973) proved there are infinitely many primes $p$ with $p+2$ a product of at most two primes.\n- The parity problem in sieve theory is a known structural obstruction to pushing $246$ down to $2$ by these methods alone.", |
| "why_it_matters": null, |
| "review_state": "published", |
| "horizon": "sharp", |
| "difficulty": "notorious", |
| "tags": [ |
| "prime-gaps", |
| "primes", |
| "sieve-theory" |
| ], |
| "posed_by": "Attributed to de Polignac (1849)", |
| "posed_year": 1849, |
| "prize_amount_usd": null, |
| "prize_sponsor": null, |
| "references": [ |
| { |
| "kind": "paper", |
| "citation": "Y. Zhang, Bounded gaps between primes, Ann. of Math. 179 (2014), 1121–1174.", |
| "url": null, |
| "note": null, |
| "sort_order": 0 |
| }, |
| { |
| "kind": "paper", |
| "citation": "J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015), 383–413.", |
| "url": "https://arxiv.org/abs/1311.4600", |
| "note": null, |
| "sort_order": 1 |
| } |
| ], |
| "provenance": { |
| "source_record": null, |
| "imported_at": null, |
| "status_as_of": null, |
| "evidence_label": null, |
| "note": null, |
| "has_full_text": null, |
| "original_file": null, |
| "sha256": null |
| }, |
| "source_status_json": null, |
| "quality_flags": [ |
| "formalization-not-compiled-during-export" |
| ], |
| "linked_problem_ids": [], |
| "dates": { |
| "created_at": "2026-09-07T23:45:31.857Z", |
| "updated_at": "2026-09-07T23:45:31.857Z", |
| "published_at": "2026-09-07T23:45:34.811Z", |
| "solved_at": null |
| }, |
| "snapshot_at": "2026-10-10T16:07:49.246Z", |
| "metadata_json": "{\"database_id\":\"e259931f-368a-4968-9963-7f38298da884\",\"format_issues\":null,\"number\":6}" |
| } |
|
|
