{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XRA5L7XQARVPUP2AZXIGGNP2P4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fd72caf6d5775a7a6ba190f8f9173db4630248ced2fe5cf8d29c0b0f743ebe4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T09:27:30Z","title_canon_sha256":"a9aad2a8c902c18c245f23ca70658beb37772c486345b445eea7386a141cf8f3"},"schema_version":"1.0","source":{"id":"1601.06361","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06361","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06361v1","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06361","created_at":"2026-05-18T01:22:04Z"},{"alias_kind":"pith_short_12","alias_value":"XRA5L7XQARVP","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XRA5L7XQARVPUP2A","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XRA5L7XQ","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:bea8625ea874d60660c6029c4125b4ca10ef63909bf283f5d29a346b64e461fb","target":"graph","created_at":"2026-05-18T01:22:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \\ne 0$ and $\\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately $0.844$ the Dirichlet density of the set of prime exponents to which the previous equation is known to not have such solutions. For the proof we develop a criterion of independent interest to decide if two elliptic curves with certain type of potentially good reduction at 2 have symplectically or anti-symplectically isomorphic $p$-torsion modules.","authors_text":"Nuno Freitas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T09:27:30Z","title":"On the Fermat-type Equation $x^3 + y^3 = z^p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06361","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:41101a25474bdf3fb8c02062d8319d04ebb5a593d34b0aecda6e7e5da4275eb2","target":"record","created_at":"2026-05-18T01:22:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fd72caf6d5775a7a6ba190f8f9173db4630248ced2fe5cf8d29c0b0f743ebe4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-24T09:27:30Z","title_canon_sha256":"a9aad2a8c902c18c245f23ca70658beb37772c486345b445eea7386a141cf8f3"},"schema_version":"1.0","source":{"id":"1601.06361","kind":"arxiv","version":1}},"canonical_sha256":"bc41d5fef0046afa3f40cdd06335fa7f2c9b949a8369334074ab1413494bda8f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc41d5fef0046afa3f40cdd06335fa7f2c9b949a8369334074ab1413494bda8f","first_computed_at":"2026-05-18T01:22:04.647399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:04.647399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G2frBRLxCHBTEJ6hgmS11t6hSn/9Fk8KtkVzo2SFKj8VQqa/MzYeXXqsg6uodvD3Xq0ININ/2TvZ9dEGxX+1Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:04.647768Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06361","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41101a25474bdf3fb8c02062d8319d04ebb5a593d34b0aecda6e7e5da4275eb2","sha256:bea8625ea874d60660c6029c4125b4ca10ef63909bf283f5d29a346b64e461fb"],"state_sha256":"865e4284b8213a2b317c2b3b37d9b0f52a73b6f184b0c485ac623cf2f04f6bfa"}