{
  "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}"
}