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We improve the upper bound, showing $\\sum_{p \\le x} A_{3}(p) \\ll x (\\log x)^3 (\\log \\log x)^2$."},"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":"2201.11805","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-01-27T20:51:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eb4bda5af879fa613e061d2e242f25d37f8752111fd25c895a729f330bf4d4aa","abstract_canon_sha256":"82728a8eb972504dddea1ceca013e1e3d5943ef9627769dea0c185faba48a35a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:58:50.218826Z","signature_b64":"HpXGuXkjrgeYL6tXDlgTxMA6COauNwd4FcD1HBJz+4ilGqh+GUp1dD5GB9+gCnKmhMM3tePebHkPNvinP44zCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41c20e9aa45f253dc37a9bb91e8e5d9bc38238aa4382f514870f82210e1e16c0","last_reissued_at":"2026-07-05T05:58:50.218477Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:58:50.218477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ternary Egyptian fractions with prime denominator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Adva Mond, Julien Portier","submitted_at":"2022-01-27T20:51:29Z","abstract_excerpt":"For a prime number $p$, let $A_3(p)= | \\{ m \\in \\mathbb{N}: \\exists m_1,m_2,m_3 \\in \\mathbb{N}, \\frac{m}{p}=\\frac{1}{m_1}+\\frac{1}{m_2}+\\frac{1}{m_3} \\} |$. In 2019 Luca and Pappalardi proved that $x (\\log x)^3 \\ll \\sum_{p \\le x} A_{3}(p) \\ll x (\\log x)^5$. We improve the upper bound, showing $\\sum_{p \\le x} A_{3}(p) \\ll x (\\log x)^3 (\\log \\log x)^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.11805","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/2201.11805/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":"2201.11805","created_at":"2026-07-05T05:58:50.218539+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.11805v3","created_at":"2026-07-05T05:58:50.218539+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.11805","created_at":"2026-07-05T05:58:50.218539+00:00"},{"alias_kind":"pith_short_12","alias_value":"IHBA5GVEL4ST","created_at":"2026-07-05T05:58:50.218539+00:00"},{"alias_kind":"pith_short_16","alias_value":"IHBA5GVEL4ST3Q32","created_at":"2026-07-05T05:58:50.218539+00:00"},{"alias_kind":"pith_short_8","alias_value":"IHBA5GVE","created_at":"2026-07-05T05:58:50.218539+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP","json":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP.json","graph_json":"https://pith.science/api/pith-number/IHBA5GVEL4ST3Q32TO4R5DS5TP/graph.json","events_json":"https://pith.science/api/pith-number/IHBA5GVEL4ST3Q32TO4R5DS5TP/events.json","paper":"https://pith.science/paper/IHBA5GVE"},"agent_actions":{"view_html":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP","download_json":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP.json","view_paper":"https://pith.science/paper/IHBA5GVE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.11805&json=true","fetch_graph":"https://pith.science/api/pith-number/IHBA5GVEL4ST3Q32TO4R5DS5TP/graph.json","fetch_events":"https://pith.science/api/pith-number/IHBA5GVEL4ST3Q32TO4R5DS5TP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP/action/storage_attestation","attest_author":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP/action/author_attestation","sign_citation":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP/action/citation_signature","submit_replication":"https://pith.science/pith/IHBA5GVEL4ST3Q32TO4R5DS5TP/action/replication_record"}},"created_at":"2026-07-05T05:58:50.218539+00:00","updated_at":"2026-07-05T05:58:50.218539+00:00"}