{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:JYMFBFZJWTLJMCOROUJPU5PINK","short_pith_number":"pith:JYMFBFZJ","schema_version":"1.0","canonical_sha256":"4e18509729b4d69609d17512fa75e86a836cf3dc92030480148b8ca21e0cc44c","source":{"kind":"arxiv","id":"1503.02798","version":2},"attestation_state":"computed","paper":{"title":"A congruence involving harmonic sums modulo $p^{\\alpha}q^{\\beta}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lirui Jia, Tianxin Cai, Zhongyan Shen","submitted_at":"2015-03-10T07:42:21Z","abstract_excerpt":"In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \\begin{equation*} Z(p^{r})\\equiv-2p^{r-1}B_{p-3} ~(\\bmod ~ p^{r}), \\end{equation*} where $ Z(n)=\\sum\\limits_{i+j+k=n\\atop{i,j,k\\in\\mathcal{P}_{n}}}\\frac{1}{ijk}$ and $\\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $\\alpha,~\\beta$, \\begin{equation*} Z(p^{\\alpha}q^{\\beta})\\equiv 2(2-q)(1-\\frac{1}{q^{3}})p^{\\alpha-1}q^{\\beta-1}B_{p-3}\\pmod{p^{\\alpha}} \\end{equation"},"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":"1503.02798","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-10T07:42:21Z","cross_cats_sorted":[],"title_canon_sha256":"1ef40a17c425a33b5de86f0aa0a2afb0a6c8c121bf3c785ad11f084c313fcb11","abstract_canon_sha256":"e4091422690bdc7a4eeed4d0ab2ca924596e4ec71bda7b7a8aac56bf1ef1103d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:36.218165Z","signature_b64":"y5qpZDh5TzssCUvGTDgCZzoJYZbUfy9Eomd43+1M+oASqNsmINTXrMwJb4iVdRvtLpBXpnjGxBZIRiR0x4WDAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e18509729b4d69609d17512fa75e86a836cf3dc92030480148b8ca21e0cc44c","last_reissued_at":"2026-05-18T00:56:36.217431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:36.217431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A congruence involving harmonic sums modulo $p^{\\alpha}q^{\\beta}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lirui Jia, Tianxin Cai, Zhongyan Shen","submitted_at":"2015-03-10T07:42:21Z","abstract_excerpt":"In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \\begin{equation*} Z(p^{r})\\equiv-2p^{r-1}B_{p-3} ~(\\bmod ~ p^{r}), \\end{equation*} where $ Z(n)=\\sum\\limits_{i+j+k=n\\atop{i,j,k\\in\\mathcal{P}_{n}}}\\frac{1}{ijk}$ and $\\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $\\alpha,~\\beta$, \\begin{equation*} Z(p^{\\alpha}q^{\\beta})\\equiv 2(2-q)(1-\\frac{1}{q^{3}})p^{\\alpha-1}q^{\\beta-1}B_{p-3}\\pmod{p^{\\alpha}} \\end{equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02798","kind":"arxiv","version":2},"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":"1503.02798","created_at":"2026-05-18T00:56:36.217554+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02798v2","created_at":"2026-05-18T00:56:36.217554+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02798","created_at":"2026-05-18T00:56:36.217554+00:00"},{"alias_kind":"pith_short_12","alias_value":"JYMFBFZJWTLJ","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JYMFBFZJWTLJMCOR","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JYMFBFZJ","created_at":"2026-05-18T12:29:27.538025+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/JYMFBFZJWTLJMCOROUJPU5PINK","json":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK.json","graph_json":"https://pith.science/api/pith-number/JYMFBFZJWTLJMCOROUJPU5PINK/graph.json","events_json":"https://pith.science/api/pith-number/JYMFBFZJWTLJMCOROUJPU5PINK/events.json","paper":"https://pith.science/paper/JYMFBFZJ"},"agent_actions":{"view_html":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK","download_json":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK.json","view_paper":"https://pith.science/paper/JYMFBFZJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02798&json=true","fetch_graph":"https://pith.science/api/pith-number/JYMFBFZJWTLJMCOROUJPU5PINK/graph.json","fetch_events":"https://pith.science/api/pith-number/JYMFBFZJWTLJMCOROUJPU5PINK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK/action/storage_attestation","attest_author":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK/action/author_attestation","sign_citation":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK/action/citation_signature","submit_replication":"https://pith.science/pith/JYMFBFZJWTLJMCOROUJPU5PINK/action/replication_record"}},"created_at":"2026-05-18T00:56:36.217554+00:00","updated_at":"2026-05-18T00:56:36.217554+00:00"}