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For any non-zero integer $y$, write $y=p_1^{k_1}\\dots p_s^{k_s}y_0$, where $k_1,\\dots,k_s$ are non-negative integers and $y_0$ is an integer coprime to $p_1,\\dots,p_s$. We define the $f$-normalized $S$-part of $y$ by $[y]_{f,S}:=p_1^{k_1 r_{p_1,S}(f)}\\dots p_s^{k_s r_{p_s,S}(f)}$, with $r_{p,S}(f)=1$ if $p\\in S\\setminus S'$ and $r_{p,S}(f)=R_{S'}(f)/R_{p}(f)$ "},"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":"1907.08239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-18T18:40:52Z","cross_cats_sorted":[],"title_canon_sha256":"3a7da207a8e27ed9ea68d1cb83442072eeacacebe15f26cbfd2c2cc4daf02521","abstract_canon_sha256":"6ebcd5f509014071676c2c877a262b2aa4b2d155e33b2196daa45d2b0cd5f008"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:10.468615Z","signature_b64":"RPB2u6fYw1vaVUrYABAtcQN/vXtnzE/nbgSJ2HA97ymQhC3qQiYJepCD8Abs+m1CkLE94FgVHBZAnIQnfguxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba3ef1ee493794973493e00da1fa0c185350fbd4ec955b274c5079d1b2dcf081","last_reissued_at":"2026-05-17T23:40:10.467953Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:10.467953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"S-parts of values of univariate polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maurizio Moreschi","submitted_at":"2019-07-18T18:40:52Z","abstract_excerpt":"Let $S=\\{p_1,\\dots,p_s\\}$ be a finite non-empty set of distinct prime numbers, let $f\\in \\mathbb{Z}[X]$ be a polynomial of degree $n\\ge 1$, and let $S'\\subseteq S$ be the subset of all $p\\in S$ such that $f$ has a root in $\\mathbb{Z}_p$. For any non-zero integer $y$, write $y=p_1^{k_1}\\dots p_s^{k_s}y_0$, where $k_1,\\dots,k_s$ are non-negative integers and $y_0$ is an integer coprime to $p_1,\\dots,p_s$. We define the $f$-normalized $S$-part of $y$ by $[y]_{f,S}:=p_1^{k_1 r_{p_1,S}(f)}\\dots p_s^{k_s r_{p_s,S}(f)}$, with $r_{p,S}(f)=1$ if $p\\in S\\setminus S'$ and $r_{p,S}(f)=R_{S'}(f)/R_{p}(f)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08239","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":"1907.08239","created_at":"2026-05-17T23:40:10.468050+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.08239v1","created_at":"2026-05-17T23:40:10.468050+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08239","created_at":"2026-05-17T23:40:10.468050+00:00"},{"alias_kind":"pith_short_12","alias_value":"XI7PD3SJG6KJ","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"XI7PD3SJG6KJONET","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"XI7PD3SJ","created_at":"2026-05-18T12:33:33.725879+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/XI7PD3SJG6KJONET4AG2D6QMDB","json":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB.json","graph_json":"https://pith.science/api/pith-number/XI7PD3SJG6KJONET4AG2D6QMDB/graph.json","events_json":"https://pith.science/api/pith-number/XI7PD3SJG6KJONET4AG2D6QMDB/events.json","paper":"https://pith.science/paper/XI7PD3SJ"},"agent_actions":{"view_html":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB","download_json":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB.json","view_paper":"https://pith.science/paper/XI7PD3SJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.08239&json=true","fetch_graph":"https://pith.science/api/pith-number/XI7PD3SJG6KJONET4AG2D6QMDB/graph.json","fetch_events":"https://pith.science/api/pith-number/XI7PD3SJG6KJONET4AG2D6QMDB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB/action/storage_attestation","attest_author":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB/action/author_attestation","sign_citation":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB/action/citation_signature","submit_replication":"https://pith.science/pith/XI7PD3SJG6KJONET4AG2D6QMDB/action/replication_record"}},"created_at":"2026-05-17T23:40:10.468050+00:00","updated_at":"2026-05-17T23:40:10.468050+00:00"}