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Consequently, for any integer $m>4$ there is a positive integer $n$ with $\\pi(mn)=m+n$. We also pose several conjectures for further research; for example, we conjecture that for each $m=1,2,3,\\ldots$ there is a positive integer $n$ such that $m+n$ divides $p_m+p_n$, where $p_k$ denotes the $k$-th"},"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":"1409.5685","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-19T14:57:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c4f23bda89bdde28ca037c9afdcf91461a3c5f56c103ca488455e71e9538a7a4","abstract_canon_sha256":"69405e24f3f0e75c3bdcaf5631e074e4e58373163edd1218d27ac046dc75443f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:10.950351Z","signature_b64":"FUVcFOHYZK0XNcUnGYmORS9Bho3zuEbOrI9e7qvUg+k/ZeeWlWlzFN8SpDUAxqi2mm/LBvMOTlvJ9igRbaDRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c5acaa892cb0ba5674ccc416d8df3d246a4aa3bacb2f131eebd2c003c1e32de","last_reissued_at":"2026-05-18T00:53:10.949724Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:10.949724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new theorem on the prime-counting function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-09-19T14:57:22Z","abstract_excerpt":"For $x>0$ let $\\pi(x)$ denote the number of primes not exceeding $x$. 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We also pose several conjectures for further research; for example, we conjecture that for each $m=1,2,3,\\ldots$ there is a positive integer $n$ such that $m+n$ divides $p_m+p_n$, where $p_k$ denotes the $k$-th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5685","kind":"arxiv","version":6},"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":"1409.5685","created_at":"2026-05-18T00:53:10.949810+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.5685v6","created_at":"2026-05-18T00:53:10.949810+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5685","created_at":"2026-05-18T00:53:10.949810+00:00"},{"alias_kind":"pith_short_12","alias_value":"NRNMVKESZMF2","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"NRNMVKESZMF2KZ2M","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"NRNMVKES","created_at":"2026-05-18T12:28:41.024544+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/NRNMVKESZMF2KZ2MZRAW3DPT2J","json":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J.json","graph_json":"https://pith.science/api/pith-number/NRNMVKESZMF2KZ2MZRAW3DPT2J/graph.json","events_json":"https://pith.science/api/pith-number/NRNMVKESZMF2KZ2MZRAW3DPT2J/events.json","paper":"https://pith.science/paper/NRNMVKES"},"agent_actions":{"view_html":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J","download_json":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J.json","view_paper":"https://pith.science/paper/NRNMVKES","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.5685&json=true","fetch_graph":"https://pith.science/api/pith-number/NRNMVKESZMF2KZ2MZRAW3DPT2J/graph.json","fetch_events":"https://pith.science/api/pith-number/NRNMVKESZMF2KZ2MZRAW3DPT2J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J/action/storage_attestation","attest_author":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J/action/author_attestation","sign_citation":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J/action/citation_signature","submit_replication":"https://pith.science/pith/NRNMVKESZMF2KZ2MZRAW3DPT2J/action/replication_record"}},"created_at":"2026-05-18T00:53:10.949810+00:00","updated_at":"2026-05-18T00:53:10.949810+00:00"}