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In this note, we prove that (i) for any odd prime power $\\ell$ and $n\\ge \\max\\{q,11-q\\}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful number; (2) for any positive odd integer $\\ell$, there exists an integer $N_{q,\\ell}$ such that for any positive integer $n\\ge N_{q,\\ell}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful num"},"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":"1706.03350","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-06-11T13:19:21Z","cross_cats_sorted":[],"title_canon_sha256":"d54db3039046d381d44fc493b5359eccf8979a8da2dfc9f6ab84fbccf0f82abd","abstract_canon_sha256":"699e14ea0d3cca19b9a0e80bd3adea8839ffb5da46b23ac97233ea17126bee93"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:37.719805Z","signature_b64":"jjSaGsosQDQZkCotIF1zs+0tiDT6V/cRO3CaZa0yVq54SitvKIStDSFbDM+38S1CJlcg9oGydfCBNjOrbh5KAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c293e6b74441bc6f0ee13f5626beb0525bad540b01b3fcd9a02a59d94364a982","last_reissued_at":"2026-05-18T00:42:37.719189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:37.719189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Powerful numbers in $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Qing-Qing Zhao, Quan-Hui Yang","submitted_at":"2017-06-11T13:19:21Z","abstract_excerpt":"Let $q$ be a positive integer. Recently, Niu and Liu proved that if $n\\ge \\max\\{q,1198-q\\}$, then the product $(1^3+q^3)(2^3+q^3)\\cdots (n^3+q^3)$ is not a powerful number. In this note, we prove that (i) for any odd prime power $\\ell$ and $n\\ge \\max\\{q,11-q\\}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful number; (2) for any positive odd integer $\\ell$, there exists an integer $N_{q,\\ell}$ such that for any positive integer $n\\ge N_{q,\\ell}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful num"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03350","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":"1706.03350","created_at":"2026-05-18T00:42:37.719274+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.03350v1","created_at":"2026-05-18T00:42:37.719274+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03350","created_at":"2026-05-18T00:42:37.719274+00:00"},{"alias_kind":"pith_short_12","alias_value":"YKJ6NN2EIG6G","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"YKJ6NN2EIG6G6DXB","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"YKJ6NN2E","created_at":"2026-05-18T12:31:56.362134+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/YKJ6NN2EIG6G6DXBH5LCNPVQKJ","json":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ.json","graph_json":"https://pith.science/api/pith-number/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/graph.json","events_json":"https://pith.science/api/pith-number/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/events.json","paper":"https://pith.science/paper/YKJ6NN2E"},"agent_actions":{"view_html":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ","download_json":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ.json","view_paper":"https://pith.science/paper/YKJ6NN2E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.03350&json=true","fetch_graph":"https://pith.science/api/pith-number/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/graph.json","fetch_events":"https://pith.science/api/pith-number/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/action/storage_attestation","attest_author":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/action/author_attestation","sign_citation":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/action/citation_signature","submit_replication":"https://pith.science/pith/YKJ6NN2EIG6G6DXBH5LCNPVQKJ/action/replication_record"}},"created_at":"2026-05-18T00:42:37.719274+00:00","updated_at":"2026-05-18T00:42:37.719274+00:00"}