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As recalled below, it is equivalent to the infinitude of primary pseudoperfect numbers. Following Butske, Jaje, and Mayernik~\\cite{ButskeJajeMayernik}, a squarefree positive integer $n$ is a \\emph{primary pseudoperfect number} if \\begin{equation}\\label{eq:ppn-def}\n  \\frac1n+\\sum_{p\\mid n}\\frac1p=1, \\end{eq"},"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":"2605.21518","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-18T12:39:12Z","cross_cats_sorted":[],"title_canon_sha256":"e8dedb60cb1bece6e14cb00a4322dd1ac8cba6f0c14a80ce9d8816bb35d847a9","abstract_canon_sha256":"80ffccdbd67a3d3b3daa93eaba1d95c43e57758675e60985ea26992415f469c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T00:02:27.472952Z","signature_b64":"cp+UIGXUxKxjnlieVfPFfMWq6AiNjuQdv6x/5Wd5Mk5VcUuF4nAydpQGHBD4c0F3Ihe4V2f31rMx4eKN9BBzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"216fd64735bdab0bf9c684356755d7017874009b828eb8c4740a9730c64fdba7","last_reissued_at":"2026-05-22T00:02:27.472452Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T00:02:27.472452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Port Fillings for Primary Pseudoperfect Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Han Wang","submitted_at":"2026-05-18T12:39:12Z","abstract_excerpt":"Erd\\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\\cdots<p_k$ and positive integers $m$ such that \\begin{equation}\\label{eq:erdos-original}\n  \\frac1{p_1}+\\cdots+\\frac1{p_k}=1-\\frac1m. \\end{equation} This is Erd\\H{o}s Problems \\#313~\\cite{ErdosProblems313}. As recalled below, it is equivalent to the infinitude of primary pseudoperfect numbers. Following Butske, Jaje, and Mayernik~\\cite{ButskeJajeMayernik}, a squarefree positive integer $n$ is a \\emph{primary pseudoperfect number} if \\begin{equation}\\label{eq:ppn-def}\n  \\frac1n+\\sum_{p\\mid n}\\frac1p=1, \\end{eq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21518","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/2605.21518/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":"2605.21518","created_at":"2026-05-22T00:02:27.472517+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.21518v1","created_at":"2026-05-22T00:02:27.472517+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21518","created_at":"2026-05-22T00:02:27.472517+00:00"},{"alias_kind":"pith_short_12","alias_value":"EFX5MRZVXWVQ","created_at":"2026-05-22T00:02:27.472517+00:00"},{"alias_kind":"pith_short_16","alias_value":"EFX5MRZVXWVQX6OG","created_at":"2026-05-22T00:02:27.472517+00:00"},{"alias_kind":"pith_short_8","alias_value":"EFX5MRZV","created_at":"2026-05-22T00:02:27.472517+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/EFX5MRZVXWVQX6OGQQ2WOVOXAF","json":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF.json","graph_json":"https://pith.science/api/pith-number/EFX5MRZVXWVQX6OGQQ2WOVOXAF/graph.json","events_json":"https://pith.science/api/pith-number/EFX5MRZVXWVQX6OGQQ2WOVOXAF/events.json","paper":"https://pith.science/paper/EFX5MRZV"},"agent_actions":{"view_html":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF","download_json":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF.json","view_paper":"https://pith.science/paper/EFX5MRZV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.21518&json=true","fetch_graph":"https://pith.science/api/pith-number/EFX5MRZVXWVQX6OGQQ2WOVOXAF/graph.json","fetch_events":"https://pith.science/api/pith-number/EFX5MRZVXWVQX6OGQQ2WOVOXAF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF/action/storage_attestation","attest_author":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF/action/author_attestation","sign_citation":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF/action/citation_signature","submit_replication":"https://pith.science/pith/EFX5MRZVXWVQX6OGQQ2WOVOXAF/action/replication_record"}},"created_at":"2026-05-22T00:02:27.472517+00:00","updated_at":"2026-05-22T00:02:27.472517+00:00"}