{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:22WNUW7M2D5DM2MWMUNMCHL5CR","short_pith_number":"pith:22WNUW7M","schema_version":"1.0","canonical_sha256":"d6acda5becd0fa366996651ac11d7d1448277dba3a22a7f43e769866d29d4ef2","source":{"kind":"arxiv","id":"2606.28718","version":1},"attestation_state":"computed","paper":{"title":"2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinping Wang, Zhao Shen","submitted_at":"2026-06-27T03:52:52Z","abstract_excerpt":"Let $T(x)=\\prod_{k=0}^{\\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\\sum_{n\\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \\[ \\nu_2\\bigl(t_5(4n+j)\\bigr)\n  =4\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,2,3\\}, \\] \\[ \\nu_2\\bigl(t_9(8n+j)\\bigr)\n  =5\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-2\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,\\ldots,7\\}. \\] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\\cite{ga} for $m=5$ and $"},"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":"2606.28718","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-27T03:52:52Z","cross_cats_sorted":[],"title_canon_sha256":"3f2df29a854835e5c93a7943c027f9657c279fb4fe3d8aa036969fd429727e60","abstract_canon_sha256":"0f433ac6e1e30655d090c8ff90cf942bd13903e4554cf9878bb6c513e8051ecb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:16:49.106775Z","signature_b64":"/vwwQknxzpFUMGVJ5XBWbUWvPrbxkQdn+w5vaFylZam28uV0X6Lecmr7UYE92l0wmyZZtfm4utPUK+nZXZD1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6acda5becd0fa366996651ac11d7d1448277dba3a22a7f43e769866d29d4ef2","last_reissued_at":"2026-06-30T01:16:49.106061Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:16:49.106061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinping Wang, Zhao Shen","submitted_at":"2026-06-27T03:52:52Z","abstract_excerpt":"Let $T(x)=\\prod_{k=0}^{\\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\\sum_{n\\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \\[ \\nu_2\\bigl(t_5(4n+j)\\bigr)\n  =4\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,2,3\\}, \\] \\[ \\nu_2\\bigl(t_9(8n+j)\\bigr)\n  =5\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-2\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,\\ldots,7\\}. \\] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\\cite{ga} for $m=5$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28718","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28718/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":"2606.28718","created_at":"2026-06-30T01:16:49.106158+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.28718v1","created_at":"2026-06-30T01:16:49.106158+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28718","created_at":"2026-06-30T01:16:49.106158+00:00"},{"alias_kind":"pith_short_12","alias_value":"22WNUW7M2D5D","created_at":"2026-06-30T01:16:49.106158+00:00"},{"alias_kind":"pith_short_16","alias_value":"22WNUW7M2D5DM2MW","created_at":"2026-06-30T01:16:49.106158+00:00"},{"alias_kind":"pith_short_8","alias_value":"22WNUW7M","created_at":"2026-06-30T01:16:49.106158+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/22WNUW7M2D5DM2MWMUNMCHL5CR","json":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR.json","graph_json":"https://pith.science/api/pith-number/22WNUW7M2D5DM2MWMUNMCHL5CR/graph.json","events_json":"https://pith.science/api/pith-number/22WNUW7M2D5DM2MWMUNMCHL5CR/events.json","paper":"https://pith.science/paper/22WNUW7M"},"agent_actions":{"view_html":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR","download_json":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR.json","view_paper":"https://pith.science/paper/22WNUW7M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.28718&json=true","fetch_graph":"https://pith.science/api/pith-number/22WNUW7M2D5DM2MWMUNMCHL5CR/graph.json","fetch_events":"https://pith.science/api/pith-number/22WNUW7M2D5DM2MWMUNMCHL5CR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR/action/storage_attestation","attest_author":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR/action/author_attestation","sign_citation":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR/action/citation_signature","submit_replication":"https://pith.science/pith/22WNUW7M2D5DM2MWMUNMCHL5CR/action/replication_record"}},"created_at":"2026-06-30T01:16:49.106158+00:00","updated_at":"2026-06-30T01:16:49.106158+00:00"}