{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:H5U5IFB2EY5ZACJ4BXHJPFCJXI","short_pith_number":"pith:H5U5IFB2","schema_version":"1.0","canonical_sha256":"3f69d4143a263b90093c0dce979449ba1976d68c500e1a4637f0d9c3992fbb1b","source":{"kind":"arxiv","id":"2210.10730","version":3},"attestation_state":"computed","paper":{"title":"Integers expressible as the sum of two rational cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Ari Shnidman (with an appendix by Ashay Burungale, Christopher Skinner), Levent Alp\\\"oge, Manjul Bhargava","submitted_at":"2022-10-19T17:14:03Z","abstract_excerpt":"We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1.\n  Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any gi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2210.10730","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-19T17:14:03Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"d4d5f900d82957e21c6750e00b34e1d8d43c776a5d601abcdd96f994b6624bf1","abstract_canon_sha256":"3b318e14219c87baab5d250ed68b3946794d193c52e560af07b82498a51feeaa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:22:36.117954Z","signature_b64":"obyEMHQWhSUhDbV9w9fCazK9T0+Xp7nBQ184EFpSMM5Kvm31rxwWMZxEO68GPds/Q/9AT57J6uth3ci8EV18DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f69d4143a263b90093c0dce979449ba1976d68c500e1a4637f0d9c3992fbb1b","last_reissued_at":"2026-07-05T09:22:36.117463Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:22:36.117463Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integers expressible as the sum of two rational cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Ari Shnidman (with an appendix by Ashay Burungale, Christopher Skinner), Levent Alp\\\"oge, Manjul Bhargava","submitted_at":"2022-10-19T17:14:03Z","abstract_excerpt":"We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1.\n  Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any gi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.10730","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.10730/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2210.10730","created_at":"2026-07-05T09:22:36.117524+00:00"},{"alias_kind":"arxiv_version","alias_value":"2210.10730v3","created_at":"2026-07-05T09:22:36.117524+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.10730","created_at":"2026-07-05T09:22:36.117524+00:00"},{"alias_kind":"pith_short_12","alias_value":"H5U5IFB2EY5Z","created_at":"2026-07-05T09:22:36.117524+00:00"},{"alias_kind":"pith_short_16","alias_value":"H5U5IFB2EY5ZACJ4","created_at":"2026-07-05T09:22:36.117524+00:00"},{"alias_kind":"pith_short_8","alias_value":"H5U5IFB2","created_at":"2026-07-05T09:22:36.117524+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":4,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.01744","citing_title":"Gross-Zagier formula for the $4, 7$ cases of Sylvester's conjecture","ref_index":92,"is_internal_anchor":false},{"citing_arxiv_id":"2606.31649","citing_title":"Tamagawa ratios and unbounded Selmer moments","ref_index":1,"is_internal_anchor":false},{"citing_arxiv_id":"2605.25917","citing_title":"A proof of the $4,7$ cases of Sylvester's conjecture on cube sums","ref_index":1,"is_internal_anchor":false},{"citing_arxiv_id":"2505.22470","citing_title":"Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two","ref_index":1,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI","json":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI.json","graph_json":"https://pith.science/api/pith-number/H5U5IFB2EY5ZACJ4BXHJPFCJXI/graph.json","events_json":"https://pith.science/api/pith-number/H5U5IFB2EY5ZACJ4BXHJPFCJXI/events.json","paper":"https://pith.science/paper/H5U5IFB2"},"agent_actions":{"view_html":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI","download_json":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI.json","view_paper":"https://pith.science/paper/H5U5IFB2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2210.10730&json=true","fetch_graph":"https://pith.science/api/pith-number/H5U5IFB2EY5ZACJ4BXHJPFCJXI/graph.json","fetch_events":"https://pith.science/api/pith-number/H5U5IFB2EY5ZACJ4BXHJPFCJXI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI/action/storage_attestation","attest_author":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI/action/author_attestation","sign_citation":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI/action/citation_signature","submit_replication":"https://pith.science/pith/H5U5IFB2EY5ZACJ4BXHJPFCJXI/action/replication_record"}},"created_at":"2026-07-05T09:22:36.117524+00:00","updated_at":"2026-07-05T09:22:36.117524+00:00"}