{"version":1,"policy":"oa-lean-v1","lean_commit":"62b6a2291302d4bbeace37642a066b7510d0145c","profile":"core","imports":["Init"],"statement_hash":"a099095daad3dc941a589fada97674f1fac034036e94a77612914e47a9b69dab","problem":{"id":9,"title":"Prove that there are infinitely many Mersenne primes","description":"Prove that there are arbitrarily large exponents p for which 2^p − 1 is prime. The statement also requires p itself to be prime, which is necessary for a Mersenne number to be prime. Natural subtraction is used, but primality and p ≥ 2 exclude the truncation edge cases. A prime-search record or probabilistic heuristic is not a proof of infinitude.\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, ∃ p : Nat, N < p ∧ prime p ∧ prime (2 ^ p - 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, ∃ p : Nat, N < p ∧ prime p ∧ prime (2 ^ p - 1))\n(fun p => 2 ≤ p ∧ ∀ d : Nat, d ∣ p → d = 1 ∨ d = p))\n\n#check OA_statement\n","proof":null}