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For $k \\in \\mathbb N$, let $\\mathcal U_k (H)$ denote the set of all $m \\in \\mathbb N$ with the following property: There exist atoms $u_1, ..., u_k, v_1, ..., v_m \\in H$ such that $u_1 \\cdot ... \\cdot u_k = v_1 \\cdot ...\\cdot v_m$. Furthermore, let $\\lambda_k (H) = \\min \\mathcal U_k (H)$ and $\\rho_k (H) = \\sup \\mathcal U_k (H)$. The sets $\\mathcal U_k (H) \\subset \\mathbb N$ are intervals which"},"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":"1503.06164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-20T17:11:26Z","cross_cats_sorted":[],"title_canon_sha256":"f3a46380e5c4a90609280fba88cabafcab2f8674b7280275360316498934bd31","abstract_canon_sha256":"0c4dc5772c1f07d849c0a8d68498be32e7fa89580f6c83ae24a4a3a99816c8fa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:48.447247Z","signature_b64":"aKknZzjE9WvBRA4d7GLvjlVeyJABo/utEGyrTBYigmTwKMsvk+fJBIq1ivqvJHAgQfTX1k+3nIr8jjcH3HdhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50306e7a3ebaf16bb59cfa8fb1e7dd3ad0c19444e25105f93274040931bb83fd","last_reissued_at":"2026-05-18T02:20:48.446423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:48.446423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On products of k atoms II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alfred Geroldinger, David J. Grynkiewicz, Pingzhi Yuan","submitted_at":"2015-03-20T17:11:26Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor (for example, rings of integers in algebraic number fields or holomorphy rings in algebraic function fields). For $k \\in \\mathbb N$, let $\\mathcal U_k (H)$ denote the set of all $m \\in \\mathbb N$ with the following property: There exist atoms $u_1, ..., u_k, v_1, ..., v_m \\in H$ such that $u_1 \\cdot ... \\cdot u_k = v_1 \\cdot ...\\cdot v_m$. Furthermore, let $\\lambda_k (H) = \\min \\mathcal U_k (H)$ and $\\rho_k (H) = \\sup \\mathcal U_k (H)$. The sets $\\mathcal U_k (H) \\subset \\mathbb N$ are intervals which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06164","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":"1503.06164","created_at":"2026-05-18T02:20:48.446562+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.06164v1","created_at":"2026-05-18T02:20:48.446562+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06164","created_at":"2026-05-18T02:20:48.446562+00:00"},{"alias_kind":"pith_short_12","alias_value":"KAYG46R6XLYW","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KAYG46R6XLYWXNM4","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KAYG46R6","created_at":"2026-05-18T12:29:27.538025+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/KAYG46R6XLYWXNM47KH3DZ65HL","json":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL.json","graph_json":"https://pith.science/api/pith-number/KAYG46R6XLYWXNM47KH3DZ65HL/graph.json","events_json":"https://pith.science/api/pith-number/KAYG46R6XLYWXNM47KH3DZ65HL/events.json","paper":"https://pith.science/paper/KAYG46R6"},"agent_actions":{"view_html":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL","download_json":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL.json","view_paper":"https://pith.science/paper/KAYG46R6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.06164&json=true","fetch_graph":"https://pith.science/api/pith-number/KAYG46R6XLYWXNM47KH3DZ65HL/graph.json","fetch_events":"https://pith.science/api/pith-number/KAYG46R6XLYWXNM47KH3DZ65HL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL/action/storage_attestation","attest_author":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL/action/author_attestation","sign_citation":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL/action/citation_signature","submit_replication":"https://pith.science/pith/KAYG46R6XLYWXNM47KH3DZ65HL/action/replication_record"}},"created_at":"2026-05-18T02:20:48.446562+00:00","updated_at":"2026-05-18T02:20:48.446562+00:00"}