{"version":1,"policy":"oa-lean-v1","lean_commit":"62b6a2291302d4bbeace37642a066b7510d0145c","profile":"core","imports":["Init"],"statement_hash":"2a31675769c3215279d404dded160281e6f1499ec9c24192095c18d553e53679","problem":{"id":8,"title":"Landau’s fourth problem: infinitely many primes of the form n² + 1","description":"Prove that for every bound N there is a natural number n > N for which n² + 1 is prime. This is a particular quadratic prime-values problem. Producing values with a bounded number of prime factors is weaker than the requested primality. The target explicitly quantifies over arbitrarily large n.\n\nResearch status checked 8 October 2026: no accepted general proof identified in the cited literature. This is a proposed formal statement; independent correspondence review is requested. A successful statement check establishes well-formedness, not the conjecture.","statement":"(fun prime : Nat → Prop =>\n  ∀ N : Nat, ∃ n : Nat, N < n ∧ prime (n ^ 2 + 1))\n(fun p => 2 ≤ p ∧ ∀ d : Nat, d ∣ p → d = 1 ∨ d = p)","profile":"core","module_pins":[],"module_context":[],"scope":{}},"source":"import Init\n\ndef OA_statement : Prop := ((fun prime : Nat → Prop =>\n  ∀ N : Nat, ∃ n : Nat, N < n ∧ prime (n ^ 2 + 1))\n(fun p => 2 ≤ p ∧ ∀ d : Nat, d ∣ p → d = 1 ∨ d = p))\n\n#check OA_statement\n","proof":null}