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In this note, we establish a combinational congruence of alternating harmonic sums for any odd prime $p$ and positive integers $r$, \\begin{equation*}\n  \\sum\\limits_{i+j+k=p^{r}\\atop{i,j,k\\in \\mathcal{P}_{p}}}\\frac{(-1)^{i}}{ijk}\n  \\equiv \\frac{1}{2}p^{r-"},"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":"1503.03156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-11T02:56:59Z","cross_cats_sorted":[],"title_canon_sha256":"db9157a804e2439129e40da9ea3d739907aea5260bb51413cd7b9a8a7f2cec15","abstract_canon_sha256":"4757f4fb5220d1db7b3beed318c06a187991abb2cded31726009e6e68775d46d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:04.203530Z","signature_b64":"51trlL4ZEjgSZuEI0Twkscxc2M8Y0WZ3GwWoFYViTqiScGNPq9LLQ1AHeRX9CLnHbmltqnNBmTwZn70B71ZGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aec3f145bd0b3f23cd9f911d4e04c056b86a58361e8ecc5ff8f6e743a4933739","last_reissued_at":"2026-05-18T02:25:04.203160Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:04.203160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Super congruences involving alternating harmonic sums modulo prime powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tianxin Cai, Zhongyan Shen","submitted_at":"2015-03-11T02:56:59Z","abstract_excerpt":"In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \\begin{equation*}\n  \\sum\\limits_{i+j+k=p^{r}\\atop{i,j,k\\in \\mathcal{P}_{p}}}\\frac{1}{ijk}\\equiv-2p^{r-1}B_{p-3}  (\\bmod   p^{r}), \\end{equation*} where $\\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we establish a combinational congruence of alternating harmonic sums for any odd prime $p$ and positive integers $r$, \\begin{equation*}\n  \\sum\\limits_{i+j+k=p^{r}\\atop{i,j,k\\in \\mathcal{P}_{p}}}\\frac{(-1)^{i}}{ijk}\n  \\equiv \\frac{1}{2}p^{r-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03156","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":"1503.03156","created_at":"2026-05-18T02:25:04.203214+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.03156v1","created_at":"2026-05-18T02:25:04.203214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.03156","created_at":"2026-05-18T02:25:04.203214+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3B7CRN5BM7S","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3B7CRN5BM7SHTM7","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3B7CRN5","created_at":"2026-05-18T12:29:44.643036+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/V3B7CRN5BM7SHTM7SEOU4BGAK2","json":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2.json","graph_json":"https://pith.science/api/pith-number/V3B7CRN5BM7SHTM7SEOU4BGAK2/graph.json","events_json":"https://pith.science/api/pith-number/V3B7CRN5BM7SHTM7SEOU4BGAK2/events.json","paper":"https://pith.science/paper/V3B7CRN5"},"agent_actions":{"view_html":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2","download_json":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2.json","view_paper":"https://pith.science/paper/V3B7CRN5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.03156&json=true","fetch_graph":"https://pith.science/api/pith-number/V3B7CRN5BM7SHTM7SEOU4BGAK2/graph.json","fetch_events":"https://pith.science/api/pith-number/V3B7CRN5BM7SHTM7SEOU4BGAK2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2/action/storage_attestation","attest_author":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2/action/author_attestation","sign_citation":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2/action/citation_signature","submit_replication":"https://pith.science/pith/V3B7CRN5BM7SHTM7SEOU4BGAK2/action/replication_record"}},"created_at":"2026-05-18T02:25:04.203214+00:00","updated_at":"2026-05-18T02:25:04.203214+00:00"}