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The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \\[\n  \\frac{1}{K(K-1)}\n  \\sum_{\\substack{1\\le a,b\\le K\\\\a\\ne b}}\n  \\prod_{p\\mid a^a-b^b}\\left(1+\\frac{\\kappa}{p}\\right)\n  \\le C_\\kappa \\] for some constant $C_\\kappa>0$, which is valid for all"},"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":"2607.03662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-04T02:31:00Z","cross_cats_sorted":[],"title_canon_sha256":"3ad06a4ba6d21ec394320ff6a5665f5852ae710177298d2e1024f7ea81eb49a0","abstract_canon_sha256":"fc61098449ce1a8bf876281e54072f7e812ca1f5e7fa799aca163cac4b1eb698"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:59.961414Z","signature_b64":"CSRFS8fBaY0xEXtYLf2u3Hk88YrQwpkH3BJFIlmr3ST2ifx0Afawul3dubrcIcB5di7fCD7C5qjhVRcbMEjnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"914c51fccdce62e3f3583bc72418eabd6aaaab2307844eccffc10c7e6caa9873","last_reissued_at":"2026-07-07T02:17:59.960615Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:59.960615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Romanoff-type theorem for $P_2$+{$a^a$: a$\\ge$ 1}","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Huixi Li, Junfeng Li, Yuchen Ding","submitted_at":"2026-07-04T02:31:00Z","abstract_excerpt":"Let $\\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\\ge$ 1:$\\Omega(m)$ $\\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \\[\n  \\frac{1}{K(K-1)}\n  \\sum_{\\substack{1\\le a,b\\le K\\\\a\\ne b}}\n  \\prod_{p\\mid a^a-b^b}\\left(1+\\frac{\\kappa}{p}\\right)\n  \\le C_\\kappa \\] for some constant $C_\\kappa>0$, which is valid for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03662","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/2607.03662/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":"2607.03662","created_at":"2026-07-07T02:17:59.960743+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.03662v1","created_at":"2026-07-07T02:17:59.960743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.03662","created_at":"2026-07-07T02:17:59.960743+00:00"},{"alias_kind":"pith_short_12","alias_value":"SFGFD7GNZZRO","created_at":"2026-07-07T02:17:59.960743+00:00"},{"alias_kind":"pith_short_16","alias_value":"SFGFD7GNZZROH42Y","created_at":"2026-07-07T02:17:59.960743+00:00"},{"alias_kind":"pith_short_8","alias_value":"SFGFD7GN","created_at":"2026-07-07T02:17:59.960743+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/SFGFD7GNZZROH42YHPDSIGHKXV","json":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV.json","graph_json":"https://pith.science/api/pith-number/SFGFD7GNZZROH42YHPDSIGHKXV/graph.json","events_json":"https://pith.science/api/pith-number/SFGFD7GNZZROH42YHPDSIGHKXV/events.json","paper":"https://pith.science/paper/SFGFD7GN"},"agent_actions":{"view_html":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV","download_json":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV.json","view_paper":"https://pith.science/paper/SFGFD7GN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.03662&json=true","fetch_graph":"https://pith.science/api/pith-number/SFGFD7GNZZROH42YHPDSIGHKXV/graph.json","fetch_events":"https://pith.science/api/pith-number/SFGFD7GNZZROH42YHPDSIGHKXV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV/action/storage_attestation","attest_author":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV/action/author_attestation","sign_citation":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV/action/citation_signature","submit_replication":"https://pith.science/pith/SFGFD7GNZZROH42YHPDSIGHKXV/action/replication_record"}},"created_at":"2026-07-07T02:17:59.960743+00:00","updated_at":"2026-07-07T02:17:59.960743+00:00"}