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(G : ℕ → ℕ),\n  (G = fun x => (Finset.filter AgohGiuga.IsStrongGiuga (Finset.Icc 1 x)).card) →\n    ∃ N O, ∀ n ≥ N, ↑(G n) ≤ O * √↑n * Real.log ↑n","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"research solved","docstring":"Let `G(X)` denote the number of exceptions `n ≤ X` to Giuga’s conjecture.\nThen for `X` larger than an absolute constant which can be made\nexplicit, `G(X) ≪ X^{1/2} log X`.\nRef: Vicentiu Tipu, _A Note on Giuga’s Conjecture_\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number 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↑n","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:41b0990e0b0d6aee30013bec0700a0512c533b9cb47a7f4290360d253a4909b4","content_root":"sha256:1f5842c233529f8dbc239ea2bb324f9f88f0e08000431fa9b46f3423703dcf5f","availability":"reference_only","row_root":"sha256:bfef0bdd8db45ddf6963ac2f5ab3de8fde88e4dc893ecc33e8340dc074cf7bc0"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.agoh_giuga.variants.isStrongGiuga_implies_isCarmichael","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.agoh_giuga.variants.isStrongGiuga_implies_isCarmichael","summary":"∀ (a : ℕ), AgohGiuga.IsStrongGiuga a → IsCarmichael a","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"research solved","docstring":"Every strong Giuga number is a Carmichael number.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ (a : ℕ), AgohGiuga.IsStrongGiuga a → IsCarmichael 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: ℕ), AgohGiuga.IsStrongGiuga a → 9 ≤ a.primeFactors.card","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"research solved","docstring":"Giuga showed that a Giuga number has at least 9 prime factors.\nRef: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ (a : ℕ), AgohGiuga.IsStrongGiuga a → 9 ≤ a.primeFactors.card","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:46b406907243162fc0131f1d2032311faabdcd1d96f14110775aa2ed2da3e608","content_root":"sha256:db2e005b930b3441f2d79be2e17eced167dcec8dbf10e2388b9105541dfde752","availability":"reference_only","row_root":"sha256:39db8b8ec1774a48a5ef4a41ee2dd13ef1ba1cbd13c85e2926e29d5e343a8c68"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.isCarmichael_561","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.isCarmichael_561","summary":"IsCarmichael 561","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"test","docstring":null,"collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"IsCarmichael 561","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:6f280f889375d99440c3e3e287c0254a7e80b1c5cb6a41a4206e8674bbaa5fc3","content_root":"sha256:1b6f3bae262bf5c32cf5b50d9344f462d4f9568f3e416f12211c55891b4e3d44","availability":"reference_only","row_root":"sha256:7d6071391a9235b4acb35a50f24032fce75730436d906bd68f78adc967b748a4"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.isStrongGiuga_iff","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.isStrongGiuga_iff","summary":"∀ {a : ℕ}, a.Composite → (AgohGiuga.IsStrongGiuga a ↔ IsCarmichael a ∧ ∃ n, ∑ p ∈ a.primeFactors, 1 / ↑p - 1 / ↑a = ↑n)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"research solved","docstring":"Giuga showed that a number `n` is strong Giuga if and only if it is\nCarmichael and `∑_{p|n} 1/p - 1/n ∈ ℕ` (i.e., if and only if it is Carmichael\nand weak Giuga).\nRef: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ {a : ℕ}, a.Composite → (AgohGiuga.IsStrongGiuga a ↔ IsCarmichael a ∧ ∃ n, ∑ p ∈ a.primeFactors, 1 / ↑p - 1 / ↑a = ↑n)","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:16132935037de91a2a82e37baebf106b09f3c6aca946f2619b985bbd9e9643c5","content_root":"sha256:1a6c64a69f6f9e9684169d530af9118ef3cca63c53fcf7c0224ccf3b7bd3daf7","availability":"reference_only","row_root":"sha256:d9bc1d295e94edddcbbc91649dfa548ad7d1fd486d2cfd9c1b9690b114eccaee"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.isWeakGiuga_iff_prime_dvd","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.isWeakGiuga_iff_prime_dvd","summary":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∀ p ∈ n.primeFactors, p ∣ n / p - 1)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"research solved","docstring":"A composite number $n$ is weak Giuga if and only if $p \\mid (\\frac{n}{p} - 1)$ for all\nprime divisors $p$ of $n$.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∀ p ∈ n.primeFactors, p ∣ n / p - 1)","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":"formal_conjectures","file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":"https://github.com/mo271/formal-conjectures/blob/2663234a28260853790aa5752d8d4550ff0ab1ca/FormalConjectures/Wikipedia/AgohGiuga.lean#L97","formal_proof_present":true,"formal_proof_sorry_free":false},"metadata_root":"sha256:3a59002e1ec0ce13502e76a4261e823ddfbabbc590bf9be0e84b54691300483f","content_root":"sha256:4d41c97d5b25b7b3a27d31684fb1b59cac1b4973bea2ac0fcb3b10e86f66bbaf","availability":"reference_only","row_root":"sha256:106dd31a6f7307bc4d9b5b181e51943517e749a8ff8ed722338902d76622010b"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.isWeakGiuga_iff_sum_primeFactors","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.isWeakGiuga_iff_sum_primeFactors","summary":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∃ m, ∑ p ∈ n.primeFactors, 1 / ↑p - 1 / ↑n = ↑m)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[\"FC100SolvedSet1\"]","category":"research solved","docstring":"A composite number $n$ is weak Giuga if and only if\n$$\n\\sum_{p\\mid n} \\frac{1}{p} - \\frac{1}{n} \\in\\mathbb{N}.\n$$\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∃ m, ∑ p ∈ n.primeFactors, 1 / ↑p - 1 / ↑n = ↑m)","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:aa641ab38c6444956bfdfffab78a8ee8a30eeeb0f7a778dde18aaedbb133b1f0","content_root":"sha256:9937cff9102f47643a701b316c8cde8b418437898bdace07d512f8aa75cd5aee","availability":"reference_only","row_root":"sha256:8ae8ce5da91418c4c5fa8643916e1266103d0db0627d1d339cefb78d6d74104f"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.korselts_criterion","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.korselts_criterion","summary":"∀ (a : ℕ), a.Composite → (IsCarmichael a ↔ Squarefree a ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ a → p - 1 ∣ a - 1)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[\"FC100SolvedSet1\"]","category":"textbook","docstring":"A composite number `a` is Carmichael if and only if it is squarefree\nand, for all prime `p` dividing `a`, we have `p - 1 ∣ a - 1`. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Textbook","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ (a : ℕ), a.Composite → (IsCarmichael a ↔ Squarefree a ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ a → p - 1 ∣ a - 1)","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:879123ab821170ebc4ddeced8cecb35925b7c8f4fbfff06cfca240b585a1558f","content_root":"sha256:196c6cabe93b3a8b14417951534fe527242157a873d9923048aff36b14f47f9e","availability":"reference_only","row_root":"sha256:a801c50f9894d0f5e7c55544656cb336a0d37ba4638f4ecf7fc9fad1ff0416bd"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgohGiuga.squarefree_of_isCarmichael","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgohGiuga.squarefree_of_isCarmichael","summary":"∀ {a : ℕ}, a.Composite → IsCarmichael a → Squarefree a","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AgohGiuga.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AgohGiuga/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AgohGiuga","subsets":"[]","category":"textbook","docstring":"A composite Carmichael number is squarefree. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Textbook","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AgohGiuga","file_first_added":"2025-03-10T12:33:13+01:00","formal_statement":"∀ {a : ℕ}, a.Composite → IsCarmichael a → Squarefree a","source_blob_root":"sha256:52e4f5900f74982f22c7925ab0d990f09c8f18f6ae4230be0d70a66a2d09a4dc","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:1da69012d18f338d79997cab814ad33d5a588ae92db98b8cc930eef35ca3c20a","content_root":"sha256:0f6b2ea13a010823e7ff0619d33cf2890562f660d7eb1222d9ffe925cb129a68","availability":"reference_only","row_root":"sha256:cf53bb581d73b8677106312ea5be801281e97d0f29c2deaceb65fad2560eb084"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgrawalConjecture.agrawal_conjecture","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgrawalConjecture.agrawal_conjecture","summary":"True ↔\n  ∀ (n r : ℕ),\n    n > 1 →\n      r > 0 →\n        n.gcd r = 1 →\n          let R := Polynomial (ZMod n);\n          have X := Polynomial.X;\n          have I := Ideal.span {X ^ r - 1};\n          (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) → Nat.Prime n ∨ ↑n ^ 2 = 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/Agrawal.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/Agrawal/"}],"metadata":{"module":"FormalConjectures.Wikipedia.Agrawal","subsets":"[]","category":"research open","docstring":"**Agrawal's Primality Conjecture.**\n\nDoes the congruence $(X-1)^n \\equiv X^n - 1 \\pmod{n, X^r-1}$ imply\n$n$ is prime (with a specific exception for $n^2 \\equiv 1 \\pmod{r}$)?\n\nWhile the \"if\" direction is a known theorem, the \"only if\" direction\nremains a conjecture.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.Agrawal","file_first_added":"2026-03-25T02:19:35+05:30","formal_statement":"True ↔\n  ∀ (n r : ℕ),\n    n > 1 →\n      r > 0 →\n        n.gcd r = 1 →\n          let R := Polynomial (ZMod n);\n          have X := Polynomial.X;\n          have I := Ideal.span {X ^ r - 1};\n          (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) → Nat.Prime n ∨ ↑n ^ 2 = 1","source_blob_root":"sha256:9b54e1f7beab92b229490e4a7c00e24d912d401dcca9f78b8f9159d925cf5bf4","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:94d515df417e7a0f2637ad47b94d7ea1857091d02967259224d0a7185456fa7f","content_root":"sha256:1c8a900b057ffd2037bfce2c2ee3ff88fed9cdb8d828ae73ff27a53877d9f755","availability":"reference_only","row_root":"sha256:9675ed55029e57b56aa19c539e3d8e95f71d2f6c847efba26a55f2259b3bb744"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AgrawalConjecture.agrawal_conjecture.variants.popovych","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AgrawalConjecture.agrawal_conjecture.variants.popovych","summary":"∀ (n r : ℕ),\n  n > 1 →\n    r > 0 →\n      n.gcd r = 1 →\n        let R := Polynomial (ZMod n);\n        have X := Polynomial.X;\n        have I := Ideal.span {X ^ r - 1};\n        (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) →\n          (Ideal.Quotient.mk I) ((X + 2) ^ n) = (Ideal.Quotient.mk I) (X ^ n + 2) → Nat.Prime n ∨ ↑n ^ 2 = 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/Agrawal.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/Agrawal/"}],"metadata":{"module":"FormalConjectures.Wikipedia.Agrawal","subsets":"[]","category":"research open","docstring":"**Roman B. Popovych Conjecture.**\nA stronger version of Agrawal's conjecture, which also considers the congruence\n$(X+2)^n \\equiv X^n + 2 \\pmod{n, X^r-1}$.\nIf both congruences hold, then $n$ is either prime or $n^2 \\equiv 1 \\pmod{r}$.\nThis variant was proposed by Roman B. Popovych in 2018.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.Agrawal","file_first_added":"2026-03-25T02:19:35+05:30","formal_statement":"∀ (n r : ℕ),\n  n > 1 →\n    r > 0 →\n      n.gcd r = 1 →\n        let R := Polynomial (ZMod n);\n        have X := Polynomial.X;\n        have I := Ideal.span {X ^ r - 1};\n        (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) →\n          (Ideal.Quotient.mk I) ((X + 2) ^ n) = (Ideal.Quotient.mk I) (X ^ n + 2) → Nat.Prime n ∨ ↑n ^ 2 = 1","source_blob_root":"sha256:9b54e1f7beab92b229490e4a7c00e24d912d401dcca9f78b8f9159d925cf5bf4","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:f382e93ee2ba9a4d473721b8f0a7daa76f4324860db1c6ce59101008376708d7","content_root":"sha256:07a01e13788f90db6e88821a66bc3e631b6d91457c2ef2c6beca325dab85a88f","availability":"reference_only","row_root":"sha256:6f759df0b8d06125d22ee54db159f8bd87599b4e5914302fa2b8bcd2e2d93028"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Algebra.exists_quadraticAlgebra_of_isQuadraticExtension","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Algebra.exists_quadraticAlgebra_of_isQuadraticExtension","summary":"∀ (K : Type u_1) (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]\n  [Algebra.IsQuadraticExtension K L], ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/WallSunSun.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/WallSunSun/"}],"metadata":{"module":"FormalConjectures.Wikipedia.WallSunSun","subsets":"[]","category":"textbook","docstring":"A quadratic algebra `L` over a field `K` is isomorphic to the explicit quadratic algebra\n`QuadraticAlgebra K a b` for some `a b : K`. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Textbook","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.WallSunSun","file_first_added":"2025-08-27T15:32:44+03:00","formal_statement":"∀ (K : Type u_1) (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]\n  [Algebra.IsQuadraticExtension K L], ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","source_blob_root":"sha256:415413e13f723d36b026d86916a4486829b5c281dd9209b8f049d3573b9183c5","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:be80fc0e93d6588e14f65536f59712b9bf9c32cbcc5119fcfba6b6c07d4b5c70","content_root":"sha256:8ee92b145e28371e918b31aab0057940c3d3bb4d4473177b9922d504224e4725","availability":"reference_only","row_root":"sha256:63b26e1c72396ab56b7dce8c2b3396cd0bebbdf2a792df9d796a282aa351d749"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Algebra.isQuadraticExtension_iff_exists_quadraticAlgebra","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Algebra.isQuadraticExtension_iff_exists_quadraticAlgebra","summary":"∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L],\n  Algebra.IsQuadraticExtension K L ↔ ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/WallSunSun.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/WallSunSun/"}],"metadata":{"module":"FormalConjectures.Wikipedia.WallSunSun","subsets":"[]","category":"textbook","docstring":"An algebra `L` is quadratic over a field `K` iff it is isomorphic to the explicit quadratic\nalgebra `QuadraticAlgebra K a b` for some `a b : K`. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Textbook","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.WallSunSun","file_first_added":"2025-08-27T15:32:44+03:00","formal_statement":"∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L],\n  Algebra.IsQuadraticExtension K L ↔ ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","source_blob_root":"sha256:415413e13f723d36b026d86916a4486829b5c281dd9209b8f049d3573b9183c5","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:6b60986bc7c15e0431ef5483e853c61a9114a48c4a9bdab8013dbbff0a66cf86","content_root":"sha256:61d1173ad487624d889b260bd239876c7b1923609ace6191fa768e3535725b4c","availability":"reference_only","row_root":"sha256:4f5dcbb1d9971d972f3cecd0b29e29c5b1802b548173cd1eb2d8df63f34b22b6"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AlmostPerfectNumbers.exists_almost_perfect_not_power_of_two","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AlmostPerfectNumbers.exists_almost_perfect_not_power_of_two","summary":"True ↔ ∃ n, AlmostPerfectNumbers.AlmostPerfect n ∧ ¬∃ k, n = 2 ^ k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AlmostPerfectNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AlmostPerfectNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AlmostPerfectNumbers","subsets":"[]","category":"research open","docstring":"**Non-Power-of-2 Almost Perfect Numbers Conjecture.**\nDoes there exist an almost perfect number that is not a power of 2?\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AlmostPerfectNumbers","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"True ↔ ∃ n, AlmostPerfectNumbers.AlmostPerfect n ∧ ¬∃ k, n = 2 ^ k","source_blob_root":"sha256:000c6f8df556e3372802fc7497094be21c35ee903a85fb90fab9b7408da9bb4d","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:3b7c3db502f62d1a8ed2783b8e3b5ce33d6b4fee9ca2da450f8957ce7db8fe6d","content_root":"sha256:81c035334e11c8ab26765b8c5b3f5befdb1281762257793c6737677cf8cc55ca","availability":"reference_only","row_root":"sha256:48e691975df3313aa7207b92929cfe07184eac9af7c84e4f8daf97ec2384b478"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AmicableNumbers.IsAmicable.symm","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AmicableNumbers.IsAmicable.symm","summary":"∀ {a b : ℕ}, IsAmicable a b → IsAmicable b a","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AmicableNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AmicableNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AmicableNumbers","subsets":"[]","category":"test","docstring":"`IsAmicable` is symmetric. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AmicableNumbers","file_first_added":"2026-03-17T08:24:48+01:00","formal_statement":"∀ {a b : ℕ}, IsAmicable a b → IsAmicable b a","source_blob_root":"sha256:d3b3b7bd9edb574e3e2738fae1b22f5b5e7c8808d3245d313b391a7c79d4b288","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:837c1f725cc5b75b2cc9546f1d8192ee21df0ab55241d8ab89e20968ddf52879","content_root":"sha256:0ceebd3daab4198d989983d6e2808beb9b81c4f470dca4a8dc4a86f95d8e04ef","availability":"reference_only","row_root":"sha256:4bc0cc551b99dd3717b870894427e3ca9656c54b840c4bee710932be4b020e30"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AmicableNumbers.amicable_220_284","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AmicableNumbers.amicable_220_284","summary":"IsAmicable 220 284","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AmicableNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AmicableNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AmicableNumbers","subsets":"[]","category":"test","docstring":"The classic amicable pair $(220, 284)$. ","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AmicableNumbers","file_first_added":"2026-03-17T08:24:48+01:00","formal_statement":"IsAmicable 220 284","source_blob_root":"sha256:d3b3b7bd9edb574e3e2738fae1b22f5b5e7c8808d3245d313b391a7c79d4b288","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:deb028d75708dcd160a817aed1cc45685b19d8dbe9dbb1ff26dab61630765aa1","content_root":"sha256:47859dce987850317c307d091074c58cf457e07cefd111b89e865de8b047cc7f","availability":"reference_only","row_root":"sha256:58dda6eba1ecedc6a43f0070fc228406a1e6919019f1f888e6fe95c2ce718190"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AmicableNumbers.infinitely_many_amicable","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AmicableNumbers.infinitely_many_amicable","summary":"sorry ↔ {(a, b) | IsAmicable a b}.Infinite","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AmicableNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AmicableNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AmicableNumbers","subsets":"[]","category":"research open","docstring":"**Infinitely many amicable numbers conjecture.**\n\nAre there infinitely many pairs of amicable numbers?\n\nWhile many amicable pairs are known, it remains open whether there are infinitely many.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers),\n[erdosproblems.com/830](https://www.erdosproblems.com/830)\n","collection":"Wikipedia","answer_kinds":"[\"Prop\"]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AmicableNumbers","file_first_added":"2026-03-17T08:24:48+01:00","formal_statement":"sorry ↔ {(a, b) | IsAmicable a b}.Infinite","source_blob_root":"sha256:d3b3b7bd9edb574e3e2738fae1b22f5b5e7c8808d3245d313b391a7c79d4b288","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:c8052cf52b11f9a2f0aedd4d0c54640c8128fe1db05524051228fd69b490c664","content_root":"sha256:f4b1bba8cfd1359f4d57b15c6dd719b46b86a18f0b4b1d72e6c90666e8b35927","availability":"reference_only","row_root":"sha256:7f958a4ff0f69df7b5542107ed1fadd121206255a27bad2805ff549684e9b60e"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AmicableNumbers.opposite_parity_amicable","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AmicableNumbers.opposite_parity_amicable","summary":"True ↔ ∃ a b, IsAmicable a b ∧ (Even a ↔ Odd b)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AmicableNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AmicableNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AmicableNumbers","subsets":"[]","category":"research open","docstring":"**Amicable numbers with opposite parity conjecture.**\nDo there exist amicable numbers $(a, b)$ where one is even and the other is odd?\n\nAll known amicable pairs are either both even or both odd. It is widely believed\nthat mixed-parity amicable pairs do not exist, but this remains open.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers)\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AmicableNumbers","file_first_added":"2026-03-17T08:24:48+01:00","formal_statement":"True ↔ ∃ a b, IsAmicable a b ∧ (Even a ↔ Odd b)","source_blob_root":"sha256:d3b3b7bd9edb574e3e2738fae1b22f5b5e7c8808d3245d313b391a7c79d4b288","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:0c8c3e2dff2b606038bef957c43de20b59a2eb17fe88d93de231d38aac79f460","content_root":"sha256:dfaf8dbbb31799c5188aafb522313e700d5e7666f76fe7dc68cdcbb39f6e7565","availability":"reference_only","row_root":"sha256:494e3d07b8bdd04c13d3e7f3f7c3b9ae72f80dbb498f1ec32e91a2720e1cdd0d"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"AmicableNumbers.relatively_prime_amicable","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"AmicableNumbers.relatively_prime_amicable","summary":"True ↔ ∃ a b, IsAmicable a b ∧ a ≠ b ∧ a.Coprime b","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/AmicableNumbers.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/AmicableNumbers/"}],"metadata":{"module":"FormalConjectures.Wikipedia.AmicableNumbers","subsets":"[]","category":"research open","docstring":"**Relatively prime amicable numbers conjecture.**\nDo there exist amicable numbers $(a, b)$ with $\\gcd(a, b) = 1$?\n\nAll known amicable pairs share a common factor. It is an open question\nwhether a pair of relatively prime amicable numbers can exist.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers)\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.AmicableNumbers","file_first_added":"2026-03-17T08:24:48+01:00","formal_statement":"True ↔ ∃ a b, IsAmicable a b ∧ a ≠ b ∧ a.Coprime b","source_blob_root":"sha256:d3b3b7bd9edb574e3e2738fae1b22f5b5e7c8808d3245d313b391a7c79d4b288","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:3a2ec25b395431ed485db39e941ba1532be2913cec55faf9959af9bea9b81880","content_root":"sha256:9a12e09453f90be99e4bbe877f6acb10caac9ae90681b806643a26b364955760","availability":"reference_only","row_root":"sha256:64cc27b792661d65a9cee87ee9393a6a9de524f202b852114c35c47843321d56"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Andrica.andrica_conjecture","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Andrica.andrica_conjecture","summary":"∀ (n : ℕ), √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/Andrica.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/Andrica/"}],"metadata":{"module":"FormalConjectures.Wikipedia.Andrica","subsets":"[]","category":"research open","docstring":"**Andrica's conjecture**\nThe inequality $\\sqrt{p_{n+1}}-\\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.Andrica","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"∀ (n : ℕ), √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","source_blob_root":"sha256:b6ea3cff22aa3107e3930b0d26c63c55b494beff91ce61230237f94d731977d6","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:0a3df775407d79a12542a660dfa857fa1d7b72158ed672544ed6d870743529c6","content_root":"sha256:5b567c1863402b931ca07e3922222362163e6e8559cd4a2e6b803b02c682e2b7","availability":"reference_only","row_root":"sha256:ee754a8738e6c5bf0f37aa0dbac222da5b43b53e9aef590c4c0110dee5dcc2f4"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Andrica.andrica_conjecture.ferreira_large_n","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Andrica.andrica_conjecture.ferreira_large_n","summary":"∀ᶠ (n : ℕ) in Filter.atTop, √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/Andrica.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/Andrica/"}],"metadata":{"module":"FormalConjectures.Wikipedia.Andrica","subsets":"[]","category":"research solved","docstring":"Ferreira proved that Andrica's conjecture is true for sufficiently large n.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.Andrica","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"∀ᶠ (n : ℕ) in Filter.atTop, √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","source_blob_root":"sha256:b6ea3cff22aa3107e3930b0d26c63c55b494beff91ce61230237f94d731977d6","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:0d45ac1205bc2cba57a340c44c8e84a0b2a0200129c85f522e468f7ebf02450f","content_root":"sha256:405b0421523586cd850c27d9970b41b8144a80a35f4b888c096e74fd0177e16f","availability":"reference_only","row_root":"sha256:326ca5b4761bb512417968b3b2a5d3d0d0b65adf729a608c2916e5e48eafcbc3"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.i","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.i","summary":"∀ (a : ℤ), ¬IsSquare a → a ≠ -1 → ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, first half.\nLet $a$ be an integer that is not a square number and not $−1$. Then the set $S(a)$\nof primes $p$ such that $a$ is a primitive root modulo $p$ has a positive asymptotic\ndensity inside the set of primes. In particular, $S(a)$ is infinite.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a : ℤ), ¬IsSquare a → a ≠ -1 → ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:4c2ad930cfdfc860d019e1b328aac68d09d29fa4e3d61c8ac06a975d6730c15a","content_root":"sha256:9dd78a2e91871be25a5cd90140476ca26c8433814d889e6126882d63c742552c","availability":"reference_only","row_root":"sha256:ad1ba16bf6f469802dfef024eef1d92079d12cfba45d7d3be4cc924206679ae4"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.ii","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.ii","summary":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half.\nWrite $a = a_0 b^2$ where $a_0$ is squarefree. Under the conditions that $a$ is not a perfect\npower and $a_0\\not\\equiv 1\\pmod{4}$ (sequence A85397 in the OEIS), the density of the set\n$S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is independent of $a$ and\nequals Artin's constant.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:362a003d52ec74db974d561520b5d680e6fad286d0fd1f8cb4af2cc5620fe0e1","content_root":"sha256:4c66a7d37b389a3cd6fa340471562cc9e020dba6ddabeee9660b907b6861963f","availability":"reference_only","row_root":"sha256:2ae6802ceda112bde7dea3b5fa94986239eb04709033261003fe09d4931d6fa1"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one","summary":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n              {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version\nIf $a = b^m$ is a perfect power of a number $b$ whose squarefree part $b_0\\equiv 1 \\pmod{4}$,\nthen the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$\nis given by\n$$C \\left(\\prod_{p \\mid m} \\frac{p(p-2)}{(p ^ 2 - p - 1)}\\right)\n\\left(1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n\\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{(1 + p - p ^ 2)}\\right),$$\nwhere $C$ is Artin's constant.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n              {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:7684cfdb7a1d8e02a72b6d3649ef7c8b20322ed055414187eea8e5f8b7db6f79","content_root":"sha256:705c366a2043804f2438241c4f95b8142e961a643060a29a35afa078402fb0e4","availability":"reference_only","row_root":"sha256:51345754262e4c0ec63d998c5a83d2c7f5225a92c6021b60d4d9f878b3ec2b1a"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one","summary":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n              {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version\nIf $a = b^m$ is a perfect odd power of a number $b$ whose squarefree part\n$b_0\\not\\equiv 1 \\pmod{4}$, then the density of the set $S(a)$ of primes $p$ such that\n$a$ is a primitive root modulo $p$ is given by\n$$C\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}$$,\nwhere $C$ is Artin's constant.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n              {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:c5b6e2e71dc13bb97b05085699a91e397a5f0d948ae23f739607eb1bb25d6d5d","content_root":"sha256:a4c481c738d8cabb625a8d2fcc00f53448515db9c058228dbc6ddcfd949fb423","availability":"reference_only","row_root":"sha256:8366e35754a4beb56f4d92b1f63cd406e307616120334fba0d298ec33361abdb"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_square_or_minus_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_square_or_minus_one","summary":"∀ (a : ℤ), IsSquare a ∨ a = -1 → (ArtinPrimitiveRootsConjecture.S a).HasDensity 0 {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, different residue version\nIf $a$ is a square number or $a = −1$, then the density of the set $S(a)$ of primes\n$p$ such that $a$ is a primitive root modulo $p$ is $0$.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a : ℤ), IsSquare a ∨ a = -1 → (ArtinPrimitiveRootsConjecture.S a).HasDensity 0 {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:ed5fe20850d136c0440fc11711074fb7d5de73a66706f6977db8e221a06fd7ea","content_root":"sha256:d4ed3a6ce2ab014ef248122aa970b7de5920dd2cd8613880af59510fe801307f","availability":"reference_only","row_root":"sha256:922732df99bbebb558d7b3b49bf89c055d1d6cb429a3cd377ac2e79fed3297d7"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.i","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.i","summary":"∀ (a : ℤ),\n  ¬IsSquare a →\n    a ≠ -1 →\n      (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n          χ.IsPrimitive →\n            ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n        ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, first half, conditional on GRH.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a : ℤ),\n  ¬IsSquare a →\n    a ≠ -1 →\n      (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n          χ.IsPrimitive →\n            ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n        ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:27d9cd1cd1a21324c71b1124bfc290fff4f38a92abb4e163f2334e8e0b61c412","content_root":"sha256:522ce8c82eabd4ad5239bcb2f1c31fd4c8b141b92e13615e49a039f4d89d7a3b","availability":"reference_only","row_root":"sha256:7deb8ca1089ccef0a27d4bb15587f35c8e1fc14f9f008acc6c446d34cb3179b2"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.ii","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.ii","summary":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n              χ.IsPrimitive →\n                ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n            (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, conditional on GRH.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n              χ.IsPrimitive →\n                ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n            (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:f9095eb9e2d71fdb6b77af7a45cb10f0edcd7d81e23d2085bfdad6fb007560e7","content_root":"sha256:4b9591e734f89f2d969505214747c2568e6055e473b2d29682dba9499adef61b","availability":"reference_only","row_root":"sha256:c8b0e96ff180db90eb33457d87ceb08642b3793b71fa583fec74b6cd5795d739"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one","summary":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                  ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n                {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH.\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                  ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n                {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:167122e79fd607cd7e6e90be2570eae6c0cdd1e0d6b17fd1d0043087805f27c5","content_root":"sha256:8611c3b525d831ebe4372f831ccb454c515a0806a77aae97c50bca4707cd7764","availability":"reference_only","row_root":"sha256:11916f03bf3ccc63beba62731341fc2425ef3ab91d1dcfa923839f1c39a1197e"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one","summary":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n                {p | Nat.Prime p}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture/"}],"metadata":{"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","subsets":"[]","category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH\n","collection":"Wikipedia","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Solved","collection_url":"https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics","display_module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","file_first_added":"2025-03-10T11:11:50+01:00","formal_statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n                {p | Nat.Prime p}","source_blob_root":"sha256:d2a8fe767b44d8d690c48dbdaa7d032bf4aeeb4dc9ac6131b59f55a9ffc27cdf","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:c1502ed5b731de49a9d416f4fcaaf6ee9c50b65d6b3d0d5ce2662cc6ffcdc3bd","content_root":"sha256:934e7626c457a9731b6812e3a41b426c5a243b0d0299bb0f177ec5195e6f505e","availability":"reference_only","row_root":"sha256:543f5c0bbf93daddc52e26bfa9338883bf49d768f760b20b2b8db779ecaeaeeb"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.id2303_01089.Tn_continuous","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.id2303_01089.Tn_continuous","summary":"∀ (n : ℕ), Continuous (Arxiv.id2303_01089.Tn n)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2303.01089%C2%BB/FurstenbergTimesPTimesQ/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","subsets":"[]","category":"API","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"28\"]","subject_names":"[\"Measure and integration\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2303.01089.FurstenbergTimesPTimesQ","file_first_added":"2026-05-07T11:13:00-04:00","formal_statement":"∀ (n : ℕ), Continuous (Arxiv.id2303_01089.Tn n)","source_blob_root":"sha256:78abd479faa4a2d45d67847da856460835be8beaf1406a10e71021b5133322b1","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:4a80a6ac7f01d3263fdf7442294e00732d7f030ee2d2e3d158a2a82922f58ac8","content_root":"sha256:f436869d738b7ddc4b441ceea0871624bdec076595e6178717cb120fc25da46d","availability":"reference_only","row_root":"sha256:ec0cf0742ee359b8c4566659d1008b518524069f84d981944f863bb5de0b2c5d"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.id2303_01089.conjecture_1_3","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.id2303_01089.conjecture_1_3","summary":"∀ {p q : ℕ},\n  2 ≤ p →\n    2 ≤ q →\n      Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n        ∀ {μ : MeasureTheory.Measure 𝕋} [MeasureTheory.IsProbabilityMeasure μ]\n          [Arxiv.id2303_01089.MeasureTheory.IsAtomLess μ],\n          MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) μ μ →\n            MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn q) μ μ → μ = MeasureTheory.volume","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2303.01089%C2%BB/FurstenbergTimesPTimesQ/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","subsets":"[]","category":"research open","docstring":"**Conjecture 1.3** (the $\\times p, \\times q$ conjecture): the only atomless Borel probability\nmeasure on $\\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"37\"]","subject_names":"[\"Dynamical systems and ergodic theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2303.01089.FurstenbergTimesPTimesQ","file_first_added":"2026-05-07T11:13:00-04:00","formal_statement":"∀ {p q : ℕ},\n  2 ≤ p →\n    2 ≤ q →\n      Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n        ∀ {μ : MeasureTheory.Measure 𝕋} [MeasureTheory.IsProbabilityMeasure μ]\n          [Arxiv.id2303_01089.MeasureTheory.IsAtomLess μ],\n          MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) μ μ →\n            MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn q) μ μ → μ = MeasureTheory.volume","source_blob_root":"sha256:78abd479faa4a2d45d67847da856460835be8beaf1406a10e71021b5133322b1","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:605fb535fc2c2b021ca00a448999ea576bfaeede6630216c72222dcaf8dd20ed","content_root":"sha256:8b62540c98ef18f8789caac67dab786c0a44ad42ca4ee57d29eccca73dfe47a9","availability":"reference_only","row_root":"sha256:dae0f92f66323b2ecf0d3c5852969ee732cd240c2c1143513557462a9d9fa5c4"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.id2303_01089.conjecture_1_4","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.id2303_01089.conjecture_1_4","summary":"False ↔\n  ∀ (p q : ℕ),\n    2 ≤ p →\n      2 ≤ q →\n        Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n          ∀ (μ : MeasureTheory.ProbabilityMeasure 𝕋),\n            Arxiv.id2303_01089.MeasureTheory.IsAtomLess ↑μ →\n              MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) ↑μ ↑μ →\n                Filter.Tendsto (fun n => μ.map ⋯) Filter.atTop\n                  (nhds Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2303.01089%C2%BB/FurstenbergTimesPTimesQ/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","subsets":"[]","category":"research solved","docstring":"**Conjecture 1.4**: if $\\mu$ is an atomless $T_p$-invariant Borel probability measure on\n$\\mathbb{T}$, then $T_{q^n}\\mu$ converges weak-star to Lebesgue measure.\nThis paper disproves the conjecture.\n","collection":"arXiv","answer_kinds":"[\"Prop\"]","subject_codes":"[\"37\"]","subject_names":"[\"Dynamical systems and ergodic theory\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2303.01089.FurstenbergTimesPTimesQ","file_first_added":"2026-05-07T11:13:00-04:00","formal_statement":"False ↔\n  ∀ (p q : ℕ),\n    2 ≤ p →\n      2 ≤ q →\n        Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n          ∀ (μ : MeasureTheory.ProbabilityMeasure 𝕋),\n            Arxiv.id2303_01089.MeasureTheory.IsAtomLess ↑μ →\n              MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) ↑μ ↑μ →\n                Filter.Tendsto (fun n => μ.map ⋯) Filter.atTop\n                  (nhds Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure)","source_blob_root":"sha256:78abd479faa4a2d45d67847da856460835be8beaf1406a10e71021b5133322b1","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:656c453df2c66b7313d1bbceb69fd952cc4d83c6f44fad02f8102bdf19cdaca7","content_root":"sha256:d6789391acb00fd6ce9236960e9bc738b9d913eab6c948d78f3f2ea1b9dcf115","availability":"reference_only","row_root":"sha256:e1901dd42785e042dc071249935c32198df8b54bd1da9ff85d69a0646f0737b7"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«0911.2077».arxiv.id0911_2077.conjecture6_3","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.0911.2077.arxiv.id0911_2077.conjecture6_3","summary":"∀ (p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ) (hk : 0 < k) (σ : ℝ),\n  σ = √(p * (1 - p)) →\n    1 - ↑(ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)) ((1 / 2 - p) * ↑(NNReal.sqrt (2 * ↑k)) / σ) +\n        1 / 2 * ↑((2 * k).choose k) * σ ^ (2 * k) ≤\n      ((PMF.binomial ⟨p, ⋯⟩ ⋯ (2 * k)).toMeasure (Set.Ici ⟨k, ⋯⟩)).toReal","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/0911.2077/Conjecture6_3.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB0911.2077%C2%BB/Conjecture6_3/"}],"metadata":{"module":"FormalConjectures.Arxiv.«0911.2077».Conjecture6_3","subsets":"[]","category":"research solved","docstring":"Empirical evidence seems to suggest that Slud's bound does not hold for all $p$, and in fact, as $n\\to\\infty$,\nthe maximal permissible $p$ shrinks to $\\frac{1}{2}$. Also, the following appears to be true:\n\nWhen $p\\in(0,1/2)$ and\n$m = 2k$ is even, and $\\sigma := \\sqrt{p(1-p)}$,\n$$\n  \\mathbb{P}[B(p,m) \\geq m/2] \\geq 1 - \\Phi\\left(\\frac{(1/2-p)\\sqrt{m}}{\\sigma}\\right) + \\frac 1 2\\binom{m}{m/2}\\sigma^{m}.\n$$\n\nA solution of this statement has been put out by Logical Intelligence\nhttps://github.com/logical-intelligence/proofs, see\n[here](https://github.com/logical-intelligence/proofs/blob/main/LI/Conj63_informal_proof.md) for\nand informal sketch of the proof.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"60\"]","subject_names":"[\"Probability theory and stochastic processes\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.0911.2077.Conjecture6_3","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"∀ (p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ) (hk : 0 < k) (σ : ℝ),\n  σ = √(p * (1 - p)) →\n    1 - ↑(ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)) ((1 / 2 - p) * ↑(NNReal.sqrt (2 * ↑k)) / σ) +\n        1 / 2 * ↑((2 * k).choose k) * σ ^ (2 * k) ≤\n      ((PMF.binomial ⟨p, ⋯⟩ ⋯ (2 * k)).toMeasure (Set.Ici ⟨k, ⋯⟩)).toReal","source_blob_root":"sha256:5a9a1498f6a1466b45b4acf0cfb407418afa62fbd8a7e7f42da24701f6b4a92e","formal_proof_kind":"lean4","file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":"https://github.com/logical-intelligence/proofs/blob/0dbb9215f472c532ca8af1376ed58a7ebca6dec2/LI/Conj63.lean#L8845","formal_proof_present":true,"formal_proof_sorry_free":false},"metadata_root":"sha256:08c3e19a3d592acce97e2c45d029f69cb1bd3df9d6b44b4f2828e21b8dbeb395","content_root":"sha256:efcb8a35e3f301c19b761db5c83d4b63e926b55fac325c2372069c3f5e42aae5","availability":"reference_only","row_root":"sha256:e0343813453e49390a86a922190a01e2c3f18b79e6861ce01f81263e11b5ceaa"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«0912.2382».curling_number_conjecture","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.0912.2382.curling_number_conjecture","summary":"∀ (S₀ : List ℤ), S₀ ≠ [] → ∃ m, Arxiv.«0912.2382».k (Arxiv.«0912.2382».S S₀ m) = 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/0912.2382/CurlingNumberConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB0912.2382%C2%BB/CurlingNumberConjecture/"}],"metadata":{"module":"FormalConjectures.Arxiv.«0912.2382».CurlingNumberConjecture","subsets":"[\"FC100OpenSet1\"]","category":"research open","docstring":"The sequence will eventually reach $1$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.0912.2382.CurlingNumberConjecture","file_first_added":"2025-03-25T20:25:32+01:00","formal_statement":"∀ (S₀ : List ℤ), S₀ ≠ [] → ∃ m, Arxiv.«0912.2382».k (Arxiv.«0912.2382».S S₀ m) = 1","source_blob_root":"sha256:ab7ca4af123bc708fdf0c95508c9404c1d9c1c623f30db9b653ae8d49903a609","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:5f9fbed715d07651f064f59077cb5c51c99d8f1b17f0d73637db18fe118f6062","content_root":"sha256:9dc33ebb5453d370761ccb12b7d31784c0c224e505b2a95da2af35419c9f0124","availability":"reference_only","row_root":"sha256:7a9f96cab543dcfa9c990bd238f905f54a31504e05217c169990ef539f24b5b1"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«1308.0994».BoxdotConjecture","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.1308.0994.BoxdotConjecture","summary":"∀ (L : Arxiv.«1308.0994».NormalModalLogic),\n  (∀ (φ : Arxiv.«1308.0994».Formula),\n      Arxiv.«1308.0994».proves L (Arxiv.«1308.0994».t φ) ↔ Arxiv.«1308.0994».proves Arxiv.«1308.0994».KT φ) →\n    L.thms ⊆ Arxiv.«1308.0994».KT.thms","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1308.0994%C2%BB/BoxdotConjecture/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1308.0994».BoxdotConjecture","subsets":"[]","category":"research solved","docstring":"Boxdot Conjecture: every normal modal logic that faithfully interprets KT\nby the boxdot translation is included in KT.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"3\"]","subject_names":"[\"Mathematical logic and foundations\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1308.0994.BoxdotConjecture","file_first_added":"2025-06-10T17:57:28-04:00","formal_statement":"∀ (L : Arxiv.«1308.0994».NormalModalLogic),\n  (∀ (φ : Arxiv.«1308.0994».Formula),\n      Arxiv.«1308.0994».proves L (Arxiv.«1308.0994».t φ) ↔ Arxiv.«1308.0994».proves Arxiv.«1308.0994».KT φ) →\n    L.thms ⊆ Arxiv.«1308.0994».KT.thms","source_blob_root":"sha256:bed2577d815f39cebf31f01889adcbe6c7e26e9e6369887055b0fbb682b9bbb8","formal_proof_kind":"lean4","file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":"https://github.com/FormalizedFormalLogic/Foundation","formal_proof_present":true,"formal_proof_sorry_free":false},"metadata_root":"sha256:15552d70ffb7013b68330bccf43a97c54482a0e3318dd506047b44efcc05a1d9","content_root":"sha256:0c30463f5f3abb8d9f513402723fad168101e19180cca4447e494564aeb1b3ff","availability":"reference_only","row_root":"sha256:b5b2287adbbc33f1d756a4deabdff01534ab19803a7a378ede40b79e3fc84199"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«1308.0994».KTExtendsK","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.1308.0994.KTExtendsK","summary":"∀ {Γ : Set Arxiv.«1308.0994».Formula} {φ : Arxiv.«1308.0994».Formula},\n  Arxiv.«1308.0994».KProof Γ φ → Arxiv.«1308.0994».KTProof Γ φ","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1308.0994/BoxdotConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1308.0994%C2%BB/BoxdotConjecture/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1308.0994».BoxdotConjecture","subsets":"[\"FC100SolvedSet1\"]","category":"API","docstring":"If `KProof Γ φ`, then `KTProof Γ φ`. In other words, KT extends K.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"3\"]","subject_names":"[\"Mathematical logic and foundations\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1308.0994.BoxdotConjecture","file_first_added":"2025-06-10T17:57:28-04:00","formal_statement":"∀ {Γ : Set Arxiv.«1308.0994».Formula} {φ : Arxiv.«1308.0994».Formula},\n  Arxiv.«1308.0994».KProof Γ φ → Arxiv.«1308.0994».KTProof Γ 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Arxiv.«1601.03081».IsCrystalWithComponents n a b → Arxiv.«1601.03081».IsCrystalWithComponents n c d → {a, b} = {c, d}","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1601.03081/UniqueCrystalComponents.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1601.03081%C2%BB/UniqueCrystalComponents/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1601.03081».UniqueCrystalComponents","subsets":"[\"FC100OpenSet1\"]","category":"research open","docstring":"If $n = ab$ is a crystal, then there are no other pairs of\npositive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and\n$B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\",\"26\"]","subject_names":"[\"Number theory\",\"Real 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Set.range a ⊆ Set.Icc 1 n) → s.length > n ^ 2 → ∃ i j, i ≠ j ∧ s[i] 0 = s[j] 0 ∧ s[i] 1 = s[j] 1","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1609.08688%C2%BB/sIncreasingrTuples/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","subsets":"[]","category":"API","docstring":"In a set of more than $n^2$ triples with coordinates from $\\{1, ..., n\\}$ we must\nhave two triples that are equal in their first two coordinates. ","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1609.08688.sIncreasingrTuples","file_first_added":"2025-03-06T12:11:55+01:00","formal_statement":"∀ {s : List (Fin 3 → ℕ)} {n : ℕ},\n  2 ≤ n → (∀ a ∈ s, Set.range a ⊆ Set.Icc 1 n) → s.length > n ^ 2 → ∃ i j, i ≠ j ∧ s[i] 0 = s[j] 0 ∧ s[i] 1 = s[j] 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: List (Fin 3 → α)},\n  Arxiv.«1609.08688».IsIncreasing₂ s → (∀ a ∈ s, ∀ (j : Fin 3), a j = val) → s.length < 2","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1609.08688%C2%BB/sIncreasingrTuples/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","subsets":"[]","category":"API","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1609.08688.sIncreasingrTuples","file_first_added":"2025-03-06T12:11:55+01:00","formal_statement":"∀ {α : Type u_1} [inst : LinearOrder α] {val : α} {s : List (Fin 3 → α)},\n  Arxiv.«1609.08688».IsIncreasing₂ s → (∀ a ∈ s, ∀ (j : Fin 3), a j = val) → s.length < 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Arxiv.«1609.08688».IsIncreasing₂ [a]","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1609.08688%C2%BB/sIncreasingrTuples/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","subsets":"[]","category":"API","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1609.08688.sIncreasingrTuples","file_first_added":"2025-03-06T12:11:55+01:00","formal_statement":"∀ {α : Type u_1} [inst : LT α] (a : Fin 3 → α), Arxiv.«1609.08688».IsIncreasing₂ 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Arxiv.«1609.08688».tripleProduct ![a, a, a] ![a, a, a] = toLex ![(a, a), (a, a), (a, a)]","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/1609.08688/sIncreasingrTuples.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB1609.08688%C2%BB/sIncreasingrTuples/"}],"metadata":{"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","subsets":"[]","category":"API","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"API","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.1609.08688.sIncreasingrTuples","file_first_added":"2025-03-06T12:11:55+01:00","formal_statement":"∀ {α : Type u_1} (a : α), Arxiv.«1609.08688».tripleProduct ![a, a, a] ![a, a, a] = toLex ![(a, a), (a, a), (a, a)]","source_blob_root":"sha256:8b411f6abd44f054094bf7c17999e698c0d3e626a22511dc889d4ae5495e855a","formal_proof_kind":null,"file_last_modified":"2026-08-02T16:46:11+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:f73c3b6021d2bc8b4e0ca1d8aeb67d067406d151b25949a6be2ae56c86aa5374","content_root":"sha256:a9d7cdedd7049dccf4fd1c4e614211b14d83b0147b31571511c8dcf25d125714","availability":"reference_only","row_root":"sha256:15bc3e2d78c27dac6c9bab7377e3620084eaa834d6c0d7f57e550d86733bcc02"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2001.02665».kotzig_conjecture_large","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2001.02665.kotzig_conjecture_large","summary":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (∀ (i : Fin (2 * n + 1)) (v : V), (f i) v = (f 0) v + i) ∧\n            (Pairwise fun i j => Disjoint (SimpleGraph.map (f i) T).edgeSet (SimpleGraph.map (f j) T).edgeSet) ∧\n              ⨆ i, SimpleGraph.map (f i) T = ⊤","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2001.02665/RingelConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2001.02665%C2%BB/RingelConjecture/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2001.02665».RingelConjecture","subsets":"[]","category":"research solved","docstring":"For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges via cyclic shifts of a single\nembedding.\n\nThe $2n+1$ copies are $f_0, f_1, \\dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices\n$v$ — that is, each copy is obtained by adding $i \\pmod{2n+1}$ to every vertex of the\nbase copy. This is strictly stronger than `ringel_conjecture_large`.\n\nKotzig conjectured this holds for all $n$; see `Paper/KotzigConjecture.lean`. Montgomery,\nPokrovskiy, and Sudakov prove it for all sufficiently large $n$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2001.02665.RingelConjecture","file_first_added":"2026-07-20T04:04:02-10:00","formal_statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (∀ (i : Fin (2 * n + 1)) (v : V), (f i) v = (f 0) v + i) ∧\n            (Pairwise fun i j => Disjoint (SimpleGraph.map (f i) T).edgeSet (SimpleGraph.map (f j) T).edgeSet) ∧\n              ⨆ i, SimpleGraph.map (f i) T = ⊤","source_blob_root":"sha256:a4281f589ede2892402bd79340582deb9bafed7fe1ba1292471b2522683ee275","formal_proof_kind":null,"file_last_modified":"2026-07-20T04:04:02-10:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:5de771d43cece342c1534522c47c16f63595dd92dc6be925aaa7483bf94020e3","content_root":"sha256:62a5480f129260c52ba23165d3b31d06a99a6cf30177f9cdf5beee7e5a9d77dc","availability":"reference_only","row_root":"sha256:73fc96e2b6d93fad6fe8c0786115343730375a278a895695c8ef17ca1e9236de"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2001.02665».ringel_conjecture_large","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2001.02665.ringel_conjecture_large","summary":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (Pairwise fun i j => Disjoint (SimpleGraph.map (f i) T).edgeSet (SimpleGraph.map (f j) T).edgeSet) ∧\n            ⨆ i, SimpleGraph.map (f i) T = ⊤","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2001.02665/RingelConjecture.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2001.02665%C2%BB/RingelConjecture/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2001.02665».RingelConjecture","subsets":"[]","category":"research solved","docstring":"For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges.\n\nA \"copy\" of $T$ is the image $T.\\text{map}(f_i)$ of $T$ under a vertex embedding\n$f_i : V \\hookrightarrow \\text{Fin}(2n+1)$; the copies are pairwise edge-disjoint\nand together cover every edge of $K_{2n+1}$.\n\nThis follows from `kotzig_conjecture_large`; see `Paper/RingelConjecture.lean` for the open form.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2001.02665.RingelConjecture","file_first_added":"2026-07-20T04:04:02-10:00","formal_statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (Pairwise fun i j => Disjoint (SimpleGraph.map (f i) T).edgeSet (SimpleGraph.map (f j) T).edgeSet) ∧\n            ⨆ i, SimpleGraph.map (f i) T = ⊤","source_blob_root":"sha256:a4281f589ede2892402bd79340582deb9bafed7fe1ba1292471b2522683ee275","formal_proof_kind":null,"file_last_modified":"2026-07-20T04:04:02-10:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:a975e07f52e85419c3eb71eb840ed0e3f967eb7760860d35821e7d700a588663","content_root":"sha256:e10c9632f97394ee025efc46d078061453bacc74fad1ca727465b26363cad8be","availability":"reference_only","row_root":"sha256:8447100eda894c7bdc57043d8af0277a0177e13015263019f1f6dc375bf531fd"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2107.00295».independentDominationEven","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2107.00295.independentDominationEven","summary":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Even G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 2) ^ 2 * i ≤ (D ^ 2 + 4) * n","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2107.00295/IndependentDomination.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2107.00295%C2%BB/IndependentDomination/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2107.00295».IndependentDomination","subsets":"[]","category":"research open","docstring":"**Conjecture 1.6 (Even case).**\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is even, then $(D + 2)^2 \\cdot i(G) \\leq (D^2 + 4) \\cdot n$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2107.00295.IndependentDomination","file_first_added":"2025-12-12T11:11:40+01:00","formal_statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Even G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 2) ^ 2 * i ≤ (D ^ 2 + 4) * n","source_blob_root":"sha256:d2b17e111aa3bd96502a0fa4a682d1d453e37ad5e90157c9a8d3cbf780c1002f","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:c8546d9489845eb28b3629aa9a9874d68e5014cefb9a731ef939e0b7b551a03b","content_root":"sha256:cf002f7967965347a0727e70722bb7d9cbcf21be6d6a420d3824c9f5736ab4fd","availability":"reference_only","row_root":"sha256:0f7c39162d72b437ef82ba60f2eabddf442ac7fd898cf7441218a4f55be760bc"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2107.00295».independentDominationOdd","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2107.00295.independentDominationOdd","summary":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Odd G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 1) * (D + 3) * i ≤ (D ^ 2 + 3) * n","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2107.00295/IndependentDomination.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2107.00295%C2%BB/IndependentDomination/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2107.00295».IndependentDomination","subsets":"[]","category":"research open","docstring":"**Conjecture 1.6 (Odd case).**\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is odd, then $(D + 1)(D + 3) \\cdot i(G) \\leq (D^2 + 3) \\cdot n$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2107.00295.IndependentDomination","file_first_added":"2025-12-12T11:11:40+01:00","formal_statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Odd G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 1) * (D + 3) * i ≤ (D ^ 2 + 3) * n","source_blob_root":"sha256:d2b17e111aa3bd96502a0fa4a682d1d453e37ad5e90157c9a8d3cbf780c1002f","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:df9d491445498e2614f3818c31dd1b399a68e3166a09ada4c84b622cfef7e0ed","content_root":"sha256:ee1a632a4e611c10ed86ae8c3bb275c13b06bd7b4004b487d14e14e3debdf1ba","availability":"reference_only","row_root":"sha256:a99654af79511c14bdd2086a0d467ebf1b0f3086e13c7790d96e7311da538553"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2107.12475».CollatzLike","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2107.12475.CollatzLike","summary":"∀ (n : ℕ), 8 < n → 2 ∈ Nat.digits 3 (2 ^ n)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2107.12475/CollatzLike.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2107.12475%C2%BB/CollatzLike/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2107.12475».CollatzLike","subsets":"[]","category":"research open","docstring":"For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.\n\nThis conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.\n\nTODO(lezeau): Formalize the Turing Machine version of this problem.\n\nSource: *Hardness of Busy Beaver Value BB(15)*: https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9\nThis is also https://arxiv.org/abs/2107.12475.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\",\"11\"]","subject_names":"[\"Combinatorics\",\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2107.12475.CollatzLike","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"∀ (n : ℕ), 8 < n → 2 ∈ Nat.digits 3 (2 ^ n)","source_blob_root":"sha256:0a1cfe2a796fc775913d673dcbcf8ec35c948ee6efca16abaeaa92de72d90704","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:e9cd5bd869b760ada8ac4f27191a016936c73a50e3896cdaa8f568a4e1f7e035","content_root":"sha256:4d801ee1d1c49559f25468ab7136ccc4ac09f76d563824e61dd6a593801bf2a8","availability":"reference_only","row_root":"sha256:fd3298a6a3537dba2fffe4dfc485527f052aed28f7fbe95028f82c08fa1f1eb3"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2107.12475».two_not_in_digits_three_pow_eight","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2107.12475.two_not_in_digits_three_pow_eight","summary":"2 ∉ Nat.digits 3 (2 ^ 8)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2107.12475/CollatzLike.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2107.12475%C2%BB/CollatzLike/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2107.12475».CollatzLike","subsets":"[]","category":"test","docstring":"For $n = 8$, $2$ is not contained in the base $3$ digits of $n$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"5\",\"11\"]","subject_names":"[\"Combinatorics\",\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2107.12475.CollatzLike","file_first_added":"2025-02-13T11:10:46+01:00","formal_statement":"2 ∉ Nat.digits 3 (2 ^ 8)","source_blob_root":"sha256:0a1cfe2a796fc775913d673dcbcf8ec35c948ee6efca16abaeaa92de72d90704","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:47ed673a50357d3b11a81207ffc9c9808effb120b16ef5e0073dbd162c405add","content_root":"sha256:579419fbd315b04f193a1ddcfb68f277f24a0a31c0fe0821effea452f69d4fb0","availability":"reference_only","row_root":"sha256:ffb4fd7f91810e8fa0748be083506052c32eed145e86d48181aeae179c798938"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2208.14736».zariski_cancellation_problem","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2208.14736.zariski_cancellation_problem","summary":"∀ {k : Type u_1} [inst : Field k] [CharZero k] {ι : Type u_2} [inst_2 : Fintype ι],\n  Arxiv.«2208.14736».IsCancellative k (MvPolynomial ι k)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2208.14736%C2%BB/ZariskiCancellation/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","subsets":"[]","category":"research open","docstring":"The **Zariski Cancellation Problem**: every polynomial ring over a field `k` of characteristic\n`0` is cancellative.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"13\",\"14\"]","subject_names":"[\"Commutative algebra\",\"Algebraic geometry\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2208.14736.ZariskiCancellation","file_first_added":"2025-02-24T11:36:04+01:00","formal_statement":"∀ {k : Type u_1} [inst : Field k] [CharZero k] {ι : Type u_2} [inst_2 : Fintype ι],\n  Arxiv.«2208.14736».IsCancellative k (MvPolynomial ι k)","source_blob_root":"sha256:9581d9406b648793288f5dba91c92d87a65faf57e198a9bfd43e022f66448335","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:c4b434f84c3e53ad79bc304330edcab9720a4d67eea1d7f10eb539d419c430ee","content_root":"sha256:208144cb5665662e52144b73bf32c68f8363a0470327bbd5f2eb14da79e07139","availability":"reference_only","row_root":"sha256:55822da1deddf5c2164522aab469a83a6e0e48fa810308ad4b7935547438fc58"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2208.14736».zariski_cancellation_problem.variants.dim_one","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2208.14736.zariski_cancellation_problem.variants.dim_one","summary":"∀ {k : Type u_1} [inst : Field k], Arxiv.«2208.14736».IsCancellative k (Polynomial k)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2208.14736%C2%BB/ZariskiCancellation/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","subsets":"[]","category":"research solved","docstring":"The single variable polynomial ring `k[X]` is cancellative in any characteristic\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"13\",\"14\"]","subject_names":"[\"Commutative algebra\",\"Algebraic 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u_1} [inst : Field k], Arxiv.«2208.14736».IsCancellative k (MvPolynomial (Fin 2) k)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2208.14736%C2%BB/ZariskiCancellation/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","subsets":"[]","category":"research solved","docstring":"The two variable polynomial ring `k[X]` is cancellative in any characteristic\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"13\",\"14\"]","subject_names":"[\"Commutative algebra\",\"Algebraic 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k)","source_blob_root":"sha256:9581d9406b648793288f5dba91c92d87a65faf57e198a9bfd43e022f66448335","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:838efb5d846a5ae239eecedba90a0ee711846e32ff328a3c66a9b20bce891bab","content_root":"sha256:cf0b1b493846e449452118407f9c10a5171c4b2efdc9980b0708cc04820cd695","availability":"reference_only","row_root":"sha256:b48121a1c9cdd792f08d848254877f0f75ff6f4a7958c276feeb22bec3b678de"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2208.14736».zariski_cancellation_problem.variants.false_pos_card","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2208.14736.zariski_cancellation_problem.variants.false_pos_card","summary":"∀ (p : ℕ) [hp : Fact (Nat.Prime p)] {ι : Type u_1} [inst : Fintype ι],\n  Fintype.card ι = 3 → ¬Arxiv.«2208.14736».IsCancellative (ZMod p) (MvPolynomial ι (ZMod p))","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2208.14736%C2%BB/ZariskiCancellation/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","subsets":"[]","category":"research solved","docstring":"The positive characteristic case of the Zariski Cancellation Problem is false in dimension `3`\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"13\",\"14\"]","subject_names":"[\"Commutative algebra\",\"Algebraic 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p))","source_blob_root":"sha256:9581d9406b648793288f5dba91c92d87a65faf57e198a9bfd43e022f66448335","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:3aa7f4492d787494d144ea7a04855bce39c52e9a9cb658de0fa0eba97cd095c7","content_root":"sha256:a3dbc2ae457bbccbbcf572f3875afffef8cca229fdd0e0193879c6fadbefac6f","availability":"reference_only","row_root":"sha256:5796a7069650ed392594524bb5eeac652bc21ae84d72d747ce37641f1c281fd6"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».S_fst_10","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.S_fst_10","summary":"List.map Arxiv.«2501.03234».S (List.range 10) = [0, 0, 1, 2, 5, 4, 7, 10, 11, 8]","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"test","docstring":"Note that in Table 1 in https://arxiv.org/abs/2501.03234v1, there seems to be an error:\n11 appears twice. The first 10 values of $S$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"List.map Arxiv.«2501.03234».S (List.range 10) = [0, 0, 1, 2, 5, 4, 7, 10, 11, 8]","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:f5ec0bec9d4ab9dc807d7ba4522bc2328e68eb52318f39d6174e94ae72245c1f","content_root":"sha256:89b7deaf30f793a7d8cd098249d3b9a3935fc332b681efb69d872133484ebf0b","availability":"reference_only","row_root":"sha256:f68d464bb89111c7c1d79ce31fe4185f3899273b323cc988556abd5956af0bfa"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_1_1","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_1_1","summary":"∀ (k : ℕ), Nat.Prime k → Odd k → 0 < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"research open","docstring":"**Conjecture 1.1**: For any odd prime $k$, the sum associated with the classical theta function $θ_3$,\n$S(k)$ is positive.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"∀ (k : ℕ), Nat.Prime k → Odd k → 0 < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:244e8203710737a0014daaa3e2c4bff0700015bb4a52608ac63c4deb1dafe028","content_root":"sha256:98b37720641f68593f9d8b14d8ccdc8de4efc0a05d14acf0b2066933d9d92a09","availability":"reference_only","row_root":"sha256:542343f7a729cceabbc39e03906b28903dee8ca3cf1018f52a383632e366f9ea"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_1","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_1","summary":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[\"FC100OpenSet1\"]","category":"research open","docstring":"**Conjecture 4.1**: For any prime $k$ larger than $5$, $S(k) > k$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:84d33fd6cf007b0d5c87c08c01763860ba1ee69fdfb89d40ba7244ff15ef5f88","content_root":"sha256:36ba2cace8ffd33ded40ba76bef008351d4c9224767fa2f426843a9d060d6180","availability":"reference_only","row_root":"sha256:28c75987c8552f23d1168e0c8981341fb85be762153705989bcb7cb0f7d4fce7"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_2","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_2","summary":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"research open","docstring":"**Conjecture 4.2**: For any prime $k$ larger than $233$, $S(k) > 2k$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:4dbed4d387f9aec902831ad161392080bedd658f82cc28e8ab32269649e6146e","content_root":"sha256:9209a5a32c2c054d97ab0ae95bc1ca8da1a1924d583c1014bfa47cb097015e94","availability":"reference_only","row_root":"sha256:8714e5dda78f7cedfd92d7e819843bdc98d1b9e53c7143df35e4a93d030dd4ec"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_3","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_3","summary":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"research open","docstring":"**Conjecture 4.3**: For any prime $k$ larger than $3119$, $S(k) > 3k$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:404bb4f779eed535603daf98fb3aac86743a3efa729bbea0746dede655957c93","content_root":"sha256:1b500b578135d7b27e6856020a0795ca0c211c382612acf93f58daeab2e9158b","availability":"reference_only","row_root":"sha256:1f34b51585ec1031b5cdf312335d4d99b253b2e46f185b92a52c797724dffb05"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_4","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_4","summary":"∀ (n : ℕ), ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑n * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"research open","docstring":"**Conjecture 4.4**: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$),\n$nk < S(k)$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Open","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"∀ (n : ℕ), ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑n * ↑k < Arxiv.«2501.03234».S 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→\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑0 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"test","docstring":"**Conjecture 1.1 → Conjecture 4.4**: If conjecture 1.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 0$. In this case the lower bound for the odd prime $k$\nwould be $0$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → 0 < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑0 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:73abdd3bcd905456e6cbcf14896b62223b1d18bec3593a9f9f32ea82c81ee296","content_root":"sha256:d721e2f84019a63ac70d787e906f5969edc3a8a27cc4d84d135855646dd7e6c0","availability":"reference_only","row_root":"sha256:2ee47098de712cdb8f391c969fc7c43e4a56c7cf46bee1e2dba46a286df948f7"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_4_def_1","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_4_def_1","summary":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑1 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"test","docstring":"**Conjecture 4.1 → Conjecture 4.4**: If conjecture 4.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 1$. In this case the lower bound would be $5$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑1 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:8377ed13c70652868a3f65dd3f87f34ea3f1c820e46592f6c8211406b4068a0d","content_root":"sha256:8f5d3a489c2ac6ee88458206cbf60eb7ec47c118850a43785b753774b95c1b32","availability":"reference_only","row_root":"sha256:72324cfb9f5a1aec335c9fe7d74ee83afc20fcf50fb6bfe4522cacfc89037f54"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_4_def_2","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_4_def_2","summary":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑2 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"test","docstring":"**Conjecture 4.2 → Conjecture 4.4**: If conjecture 4.2 holds true, then this implies a special\ncase of conjecture 4.4 for $n = 2$. For this scenario, the lower bound is now $233$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑2 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:d0ece8c8381c5b4a898f83c5e3b0110233de8c64b364d9ea5e32d165740ce32d","content_root":"sha256:aded4e793eddc0d34b5ec9190505fe3f4a7a1de557e9eec8214a15cf82e50c91","availability":"reference_only","row_root":"sha256:b4eb84e9228974712422afa668483c08d6414c1a8db0eab84f8ed871c65f3759"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2501.03234».conjecture_4_4_def_3","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2501.03234.conjecture_4_4_def_3","summary":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑3 * ↑k < Arxiv.«2501.03234».S k","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2501.03234/ArithmeticSumS.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2501.03234%C2%BB/ArithmeticSumS/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","subsets":"[]","category":"test","docstring":"**Conjecture 4.3 → Conjecture 4.4**: If conjecture 4.3 holds true, then a special\ncase of conjecture 4.4 for $n = 3$ is obtained, and the lower bound is $3119$.\n","collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"11\"]","subject_names":"[\"Number theory\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2501.03234.ArithmeticSumS","file_first_added":"2025-08-28T18:58:06+10:00","formal_statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑3 * ↑k < Arxiv.«2501.03234».S k","source_blob_root":"sha256:da8e3a0f039e83dd6b47019deadca97b0cf3f94e71b84082dc2364762d40ba3c","formal_proof_kind":null,"file_last_modified":"2026-07-17T01:28:26+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:c5fe46eeddfdea435e597dc45c74e37bc69f2fbd253534e79a2c4e780810abcd","content_root":"sha256:ce08fe5c22bd628a430a50d2c4b82b96e421223569dad7ee082af8a11e361363","availability":"reference_only","row_root":"sha256:af9e0af7929bcf0a48bf8984c4f5fb0a632d58a9422c3df1a0768a1ce11c03da"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2602.05192».epsilon_light_subset_exists","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2602.05192.epsilon_light_subset_exists","summary":"True ↔\n  ∃ c > 0,\n    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (ε : ℝ),\n      0 < ε → ε < 1 → ∃ S, Arxiv.«2602.05192».IsEpsilonLight G ε S ∧ ↑S.card ≥ c * ε * ↑n","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2602.05192/FirstProof6.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2602.05192%C2%BB/FirstProof6/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof6","subsets":"[]","category":"research solved","docstring":"Does there exist a constant $c > 0$ so that for every graph $G$ and every $\\epsilon$ between\n$0$ and $1$, $V$ contains an $\\epsilon$-light subset $S$ of size at least $c \\epsilon |V|$?\n\n","collection":"arXiv","answer_kinds":"[\"Prop\"]","subject_codes":"[\"5\"]","subject_names":"[\"Combinatorics\"]","category_label":"Solved","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2602.05192.FirstProof6","file_first_added":"2026-02-11T11:47:51+01:00","formal_statement":"True ↔\n  ∃ c > 0,\n    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (ε : ℝ),\n      0 < ε → ε < 1 → ∃ S, Arxiv.«2602.05192».IsEpsilonLight G ε S ∧ ↑S.card ≥ c * ε * ↑n","source_blob_root":"sha256:6124fa3de07b6dcb68598ea6fa7baccb014ce4116744698127e641315f27427c","formal_proof_kind":"lean4","file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":"https://github.com/frenzymath/Archon-FirstProof-Results/blob/main/FirstProof/FirstProof6/Problem6.lean","formal_proof_present":true,"formal_proof_sorry_free":false},"metadata_root":"sha256:fb7bbdea690c12c274051cb5b66256cd5818b7ddbaab0e774b7c6b212010890a","content_root":"sha256:885d06d9d9ae06b554df5751eec547fb21f82da491477426cbe4b15786d40b00","availability":"reference_only","row_root":"sha256:2216a12ceb78b4ce7e1fdb74db4087d9ebfc1bbec0f2d9bdeba12419ad486a73"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2602.05192».finiteAdditiveConvolution_comm","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2602.05192.finiteAdditiveConvolution_comm","summary":"∀ {F : Type} [inst : Field F] (n : ℕ) (p q : Polynomial F),\n  Arxiv.«2602.05192».finiteAdditiveConvolution n p q = Arxiv.«2602.05192».finiteAdditiveConvolution n q p","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2602.05192/FirstProof4.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2602.05192%C2%BB/FirstProof4/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","subsets":"[]","category":"test","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"26\"]","subject_names":"[\"Real functions\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2602.05192.FirstProof4","file_first_added":"2026-02-13T07:43:12+01:00","formal_statement":"∀ {F : Type} [inst : Field F] (n : ℕ) (p q : Polynomial F),\n  Arxiv.«2602.05192».finiteAdditiveConvolution n p q = Arxiv.«2602.05192».finiteAdditiveConvolution n q p","source_blob_root":"sha256:99fdffce0be3963d1a2b2f136e123a4aa446ac3d8815c646eae8c18c690c1fe0","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":true},"metadata_root":"sha256:961713a10bdc29f83a229e6837766b8217b35e392038a0d0609a1505f2203a50","content_root":"sha256:4f6e5ad169e01fbc5bd03ea694c5e3c4c178fb0b09bb12a2791823ef9b6d1842","availability":"reference_only","row_root":"sha256:1e8ebb6c0b2ea1bee8e5475c4c211363ff14fe2d01ba4ed1cee7db592221230d"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2602.05192».finiteAdditiveConvolution_degree","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2602.05192.finiteAdditiveConvolution_degree","summary":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  p.degree = ↑n → q.degree = ↑n → (Arxiv.«2602.05192».finiteAdditiveConvolution n p q).degree = ↑n","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2602.05192/FirstProof4.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2602.05192%C2%BB/FirstProof4/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","subsets":"[]","category":"test","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"26\"]","subject_names":"[\"Real functions\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2602.05192.FirstProof4","file_first_added":"2026-02-13T07:43:12+01:00","formal_statement":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  p.degree = ↑n → q.degree = ↑n → (Arxiv.«2602.05192».finiteAdditiveConvolution n p q).degree = ↑n","source_blob_root":"sha256:99fdffce0be3963d1a2b2f136e123a4aa446ac3d8815c646eae8c18c690c1fe0","formal_proof_kind":null,"file_last_modified":"2026-08-02T18:22:25+02:00","formal_proof_locator":null,"formal_proof_present":false,"formal_proof_sorry_free":false},"metadata_root":"sha256:2c91280e5a461e58d2f00717a9e4dbb4da627d6efe2b2a9034c4ea38a789b528","content_root":"sha256:9ac7a85b55c6c8e455141a19b5c4124a86e8a3fceb4583cdfb2dfa057eda7e6d","availability":"reference_only","row_root":"sha256:70654e951f64a9f6bbbdaeb95a61a00b9a8549ddbdb5ecbdb0302a2232809d90"},{"schema":"vela.math-native-record.v1","source_id":"source:formal-conjectures","observation_root":"sha256:a10796d13044ef5f9700b43b804c4c7ed4f326ad1bd8e204933baa3b53517f2e","native_id":"Arxiv.«2602.05192».finiteAdditiveConvolution_monic'","native_kind":"formal_conjecture","native_revision":"59f30aa314ba225fcd9268723ce8291616df1ab0","title":"Arxiv.2602.05192.finiteAdditiveConvolution_monic'","summary":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  0 < n → p.degree = ↑n → q.degree = ↑n → p.Monic → q.Monic → (Arxiv.«2602.05192».finiteAdditiveConvolution n p q).Monic","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/google-deepmind/formal-conjectures/blob/59f30aa314ba225fcd9268723ce8291616df1ab0/FormalConjectures/Arxiv/2602.05192/FirstProof4.lean"},{"locator_id":"native-2","kind":"artifact","url":"https://google-deepmind.github.io/src/FormalConjectures/Arxiv/%C2%AB2602.05192%C2%BB/FirstProof4/"}],"metadata":{"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","subsets":"[\"FC100SolvedSet1\"]","category":"test","docstring":null,"collection":"arXiv","answer_kinds":"[]","subject_codes":"[\"26\"]","subject_names":"[\"Real functions\"]","category_label":"Test","collection_url":"https://arxiv.org/archive/math","display_module":"FormalConjectures.Arxiv.2602.05192.FirstProof4","file_first_added":"2026-02-13T07:43:12+01:00","formal_statement":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  0 < n → p.degree = ↑n → 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