{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:PPXOS2URVMOOUTSHAPY53IDB57","short_pith_number":"pith:PPXOS2UR","schema_version":"1.0","canonical_sha256":"7beee96a91ab1cea4e4703f1dda061efcdf10166299402126261b79d69010d27","source":{"kind":"arxiv","id":"1507.03101","version":1},"attestation_state":"computed","paper":{"title":"Proof of a Conjecture on 6-colored Generalized Frobenius Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2015-07-11T11:50:31Z","abstract_excerpt":"Let $c\\phi_{k}(n)$ be the $k$-colored generalized Frobenius partition function. By employing the generating function of $c\\phi_{6}(3n+1)$ found by Hirschhorn, we prove that $c\\phi_{6}(27n+16)\\equiv 0$ (mod 243). This confirms a conjecture of E.X.W. Xia. We also find a congruence relation $c\\phi_{6}(81n+61) \\equiv 3 c\\phi_{6}(9n+7)$ (mod 243). Moreover, we show that $c\\phi_{6}(81n+61) \\equiv 0$ (mod 81), $c\\phi_{6}(243n+142) \\equiv 0$ (mod 243) and $c\\phi_{6}(729n+ 547) \\equiv 0$ (mod 243). We further conjecture that for $n\\ge 0$, $c\\phi_{6}(243n+142) \\equiv 0$ (mod 729)."},"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":"1507.03101","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-11T11:50:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"573539b2cf94ede42e32b32587da0fcc20b702eceb557493fb8ebc36dd7a243f","abstract_canon_sha256":"a3ad940fa0cac9381b72c6b2434444382a6c940d4d46a9ea2036adf85bfbba7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:59.686967Z","signature_b64":"ZHKe5ah0MJC21Ya9gkyhBSr+yMJ3SE0lQvPeXlacf9YEBbbFGJpWgUz4E75plcP0uW9v5YUt8gJHIlEo7/0TAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7beee96a91ab1cea4e4703f1dda061efcdf10166299402126261b79d69010d27","last_reissued_at":"2026-05-18T01:36:59.686462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:59.686462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a Conjecture on 6-colored Generalized Frobenius Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2015-07-11T11:50:31Z","abstract_excerpt":"Let $c\\phi_{k}(n)$ be the $k$-colored generalized Frobenius partition function. By employing the generating function of $c\\phi_{6}(3n+1)$ found by Hirschhorn, we prove that $c\\phi_{6}(27n+16)\\equiv 0$ (mod 243). This confirms a conjecture of E.X.W. Xia. We also find a congruence relation $c\\phi_{6}(81n+61) \\equiv 3 c\\phi_{6}(9n+7)$ (mod 243). Moreover, we show that $c\\phi_{6}(81n+61) \\equiv 0$ (mod 81), $c\\phi_{6}(243n+142) \\equiv 0$ (mod 243) and $c\\phi_{6}(729n+ 547) \\equiv 0$ (mod 243). We further conjecture that for $n\\ge 0$, $c\\phi_{6}(243n+142) \\equiv 0$ (mod 729)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03101","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1507.03101","created_at":"2026-05-18T01:36:59.686558+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.03101v1","created_at":"2026-05-18T01:36:59.686558+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03101","created_at":"2026-05-18T01:36:59.686558+00:00"},{"alias_kind":"pith_short_12","alias_value":"PPXOS2URVMOO","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"PPXOS2URVMOOUTSH","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"PPXOS2UR","created_at":"2026-05-18T12:29:37.295048+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/PPXOS2URVMOOUTSHAPY53IDB57","json":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57.json","graph_json":"https://pith.science/api/pith-number/PPXOS2URVMOOUTSHAPY53IDB57/graph.json","events_json":"https://pith.science/api/pith-number/PPXOS2URVMOOUTSHAPY53IDB57/events.json","paper":"https://pith.science/paper/PPXOS2UR"},"agent_actions":{"view_html":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57","download_json":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57.json","view_paper":"https://pith.science/paper/PPXOS2UR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.03101&json=true","fetch_graph":"https://pith.science/api/pith-number/PPXOS2URVMOOUTSHAPY53IDB57/graph.json","fetch_events":"https://pith.science/api/pith-number/PPXOS2URVMOOUTSHAPY53IDB57/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57/action/storage_attestation","attest_author":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57/action/author_attestation","sign_citation":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57/action/citation_signature","submit_replication":"https://pith.science/pith/PPXOS2URVMOOUTSHAPY53IDB57/action/replication_record"}},"created_at":"2026-05-18T01:36:59.686558+00:00","updated_at":"2026-05-18T01:36:59.686558+00:00"}