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"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}"
}
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