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In this paper, by the vanishing property given by Hou et al, we show an infinite family of congruence for $b_{11}(n)$ modulo $11$. Moreover, for $\\ell= 3, 13$ and $25$, we obtain three infinite families of congruences for $b_{\\ell}(n)$ modulo $3, 5$ and $13$ by the theory of Hecke eigenforms."},"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":"1509.07591","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T05:42:40Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"22458bd0a1824855915c584905b3eae2fbe3cd6550cd10f5800eaa4024e5aed3","abstract_canon_sha256":"021c456035dd5645ebb75a63d1d3de88340fc51688bc68c98f987ed0fb53655a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:02.411964Z","signature_b64":"GHPpQtrX1roceLR/DPM2amZk8+U0wMv9zjSvoxUGHHBzfD7Lbbu7OzL2WbyTtoyuIe4bfTWdZjVDET79X5S+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"479bdcc9af2c3cc67f6a95f4bcb0e9b252647dd1f7dab2aec76f174a3cdf352d","last_reissued_at":"2026-05-18T01:32:02.411440Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:02.411440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ramanujan-type Congruences for $\\ell$-Regular Partitions Modulo $3, 5, 11$ and $13$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hai-Tao Jin, Li Zhang","submitted_at":"2015-09-25T05:42:40Z","abstract_excerpt":"Let $b_\\ell(n)$ be the number of $\\ell$-regular partitions of $n$. 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Moreover, for $\\ell= 3, 13$ and $25$, we obtain three infinite families of congruences for $b_{\\ell}(n)$ modulo $3, 5$ and $13$ by the theory of Hecke eigenforms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07591","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":"1509.07591","created_at":"2026-05-18T01:32:02.411545+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07591v1","created_at":"2026-05-18T01:32:02.411545+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07591","created_at":"2026-05-18T01:32:02.411545+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6N5ZSNPFQ6M","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6N5ZSNPFQ6MM73K","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6N5ZSNP","created_at":"2026-05-18T12:29:25.134429+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/I6N5ZSNPFQ6MM73KSX2LZMHJWJ","json":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ.json","graph_json":"https://pith.science/api/pith-number/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/graph.json","events_json":"https://pith.science/api/pith-number/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/events.json","paper":"https://pith.science/paper/I6N5ZSNP"},"agent_actions":{"view_html":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ","download_json":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ.json","view_paper":"https://pith.science/paper/I6N5ZSNP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07591&json=true","fetch_graph":"https://pith.science/api/pith-number/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/graph.json","fetch_events":"https://pith.science/api/pith-number/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/action/storage_attestation","attest_author":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/action/author_attestation","sign_citation":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/action/citation_signature","submit_replication":"https://pith.science/pith/I6N5ZSNPFQ6MM73KSX2LZMHJWJ/action/replication_record"}},"created_at":"2026-05-18T01:32:02.411545+00:00","updated_at":"2026-05-18T01:32:02.411545+00:00"}