{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MH65JN5KRK5YUFGI5FD7QDBBTM","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":"b08a924237ff89a4960704c10d18c26e040236419998279ed682d0bda6cc50db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-05T10:58:53Z","title_canon_sha256":"015a26a22d40f6c54539fbeb7eeaee53235fffe3dd798260cc6d9d532aa5e2c3"},"schema_version":"1.0","source":{"id":"1412.1950","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.1950","created_at":"2026-05-18T02:32:05Z"},{"alias_kind":"arxiv_version","alias_value":"1412.1950v1","created_at":"2026-05-18T02:32:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1950","created_at":"2026-05-18T02:32:05Z"},{"alias_kind":"pith_short_12","alias_value":"MH65JN5KRK5Y","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MH65JN5KRK5YUFGI","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MH65JN5K","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:77066694fab37e2675c76f4ceb543db0eca5082476fe31f10fe29914f92cafa7","target":"graph","created_at":"2026-05-18T02:32:05Z","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":"A nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\\in \\mathbb{Q}^\\times$. In this paper, we prove that for any odd integer $k\\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We give also a general construction of Heegner point and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curve related to the cube sum problem.","authors_text":"Jie Shu, Li Cai, Ye Tian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-05T10:58:53Z","title":"Cube Sum Problem and an Explicit Gross-Zagier Formula"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1950","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":"sha256:8bcbb73d0b382796eff1400abf52e02659e25f34cc3ad20deabb220c292edda5","target":"record","created_at":"2026-05-18T02:32:05Z","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":"b08a924237ff89a4960704c10d18c26e040236419998279ed682d0bda6cc50db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-05T10:58:53Z","title_canon_sha256":"015a26a22d40f6c54539fbeb7eeaee53235fffe3dd798260cc6d9d532aa5e2c3"},"schema_version":"1.0","source":{"id":"1412.1950","kind":"arxiv","version":1}},"canonical_sha256":"61fdd4b7aa8abb8a14c8e947f80c219b0455163c26b1fd8e80dbe46d78e92ccd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61fdd4b7aa8abb8a14c8e947f80c219b0455163c26b1fd8e80dbe46d78e92ccd","first_computed_at":"2026-05-18T02:32:05.729344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:05.729344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LZjmE9XG+LzHJQnk05LtIq38j5kTpM0YCMvj2yTvfuUT8SCSjLLRly5hZ2yzB13ekLfirJa4Ik9oOQtCCvdPCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:05.729709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.1950","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8bcbb73d0b382796eff1400abf52e02659e25f34cc3ad20deabb220c292edda5","sha256:77066694fab37e2675c76f4ceb543db0eca5082476fe31f10fe29914f92cafa7"],"state_sha256":"32cb9ffcdb0cecfb470325070dc580e3136d962a0ef38deb9f870639a5e47378"}