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In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\\sum_{n=0}^{\\infty} a_{n}q^{n} = \\frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\\infty}^2},$$ we determine $\\alpha = \\alpha(m,n,p)$ such that $a_{pt+ \\alpha}=0$. Our results are proven using involutive transformations on integer lattices."},"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":"2606.06701","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-04T20:40:37Z","cross_cats_sorted":[],"title_canon_sha256":"2bee6e09a18064fa20da4b3f79070c67754eb8e4134b33921c0c3d0d5cd89f4f","abstract_canon_sha256":"00f56985dd1aeb5dda32b57088d9e986f1c5f22ae977fa1abb735b9692ef2294"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:23.494247Z","signature_b64":"kgSyINf8q3rHj4RTyvQp/rFQfJELqYOvtIzvh1UKZrjJKcWS3e8f575puCkzTUGvQJLbM1RHILf+SovII/XJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d937f45ef39a259b11941e34fde47e0efccbc194ff97d3dde68a1d3e548d461d","last_reissued_at":"2026-06-08T01:04:23.493356Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:23.493356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vanishing Coefficients in Products of Quintuple Products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dongxi Ye, James McLaughlin, Taylor Daniels, Tim Huber","submitted_at":"2026-06-04T20:40:37Z","abstract_excerpt":"Explicit arithmetic progressions modulo primes $p \\equiv 1 \\pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\\sum_{n=0}^{\\infty} a_{n}q^{n} = \\frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\\infty}^2},$$ we determine $\\alpha = \\alpha(m,n,p)$ such that $a_{pt+ \\alpha}=0$. Our results are proven using involutive transformations on integer lattices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06701","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/2606.06701/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":"2606.06701","created_at":"2026-06-08T01:04:23.493514+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.06701v1","created_at":"2026-06-08T01:04:23.493514+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06701","created_at":"2026-06-08T01:04:23.493514+00:00"},{"alias_kind":"pith_short_12","alias_value":"3E37IXXTTISZ","created_at":"2026-06-08T01:04:23.493514+00:00"},{"alias_kind":"pith_short_16","alias_value":"3E37IXXTTISZWEMU","created_at":"2026-06-08T01:04:23.493514+00:00"},{"alias_kind":"pith_short_8","alias_value":"3E37IXXT","created_at":"2026-06-08T01:04:23.493514+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/3E37IXXTTISZWEMUDY2P3ZD6B3","json":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3.json","graph_json":"https://pith.science/api/pith-number/3E37IXXTTISZWEMUDY2P3ZD6B3/graph.json","events_json":"https://pith.science/api/pith-number/3E37IXXTTISZWEMUDY2P3ZD6B3/events.json","paper":"https://pith.science/paper/3E37IXXT"},"agent_actions":{"view_html":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3","download_json":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3.json","view_paper":"https://pith.science/paper/3E37IXXT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.06701&json=true","fetch_graph":"https://pith.science/api/pith-number/3E37IXXTTISZWEMUDY2P3ZD6B3/graph.json","fetch_events":"https://pith.science/api/pith-number/3E37IXXTTISZWEMUDY2P3ZD6B3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3/action/storage_attestation","attest_author":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3/action/author_attestation","sign_citation":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3/action/citation_signature","submit_replication":"https://pith.science/pith/3E37IXXTTISZWEMUDY2P3ZD6B3/action/replication_record"}},"created_at":"2026-06-08T01:04:23.493514+00:00","updated_at":"2026-06-08T01:04:23.493514+00:00"}