{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ZTOFIJY4B3AKVSRBVQ4FFHA3LR","short_pith_number":"pith:ZTOFIJY4","schema_version":"1.0","canonical_sha256":"ccdc54271c0ec0aaca21ac38529c1b5c4ef4fae27b2e6c4b36b8bb00cf2612dd","source":{"kind":"arxiv","id":"1908.05220","version":1},"attestation_state":"computed","paper":{"title":"Maximally additively reducible subsets of the integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Gal Gross","submitted_at":"2019-08-14T16:45:32Z","abstract_excerpt":"Let $A, B \\subseteq \\mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \\subseteq \\mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\\{0, 1, \\ldots, k\\}$ which have $0$ as an element, the full interval $\\{0, 1, \\ldots,k\\}$ has the most divisors. To generalize to sets which do not have $0$ as an element, we prove a correspondence between additive divisors and lunar multiplication, introduced by Appelgate, LeBrun and Sloane (2011) in their study of a kind of min/max arithmetic. The number of binary lunar divisor"},"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":"1908.05220","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-14T16:45:32Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7b146f8f9c26ab089cf97c5f2fc50a14c66ac1e8f70d5d957471761005d8b9f4","abstract_canon_sha256":"74c34e9c31525a9336326fbb7bad6e9c7130d83b7e17888308aaf78107e3bf4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:09:47.744015Z","signature_b64":"f+NJb6/iQ/UOD5n3GGFHfEcsNuuxOJy4CCE39HOtHxwg5XigN1equhyploAAilzOzMRNDkbati5tV9VCwfVECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccdc54271c0ec0aaca21ac38529c1b5c4ef4fae27b2e6c4b36b8bb00cf2612dd","last_reissued_at":"2026-07-05T09:09:47.743571Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:09:47.743571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximally additively reducible subsets of the integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Gal Gross","submitted_at":"2019-08-14T16:45:32Z","abstract_excerpt":"Let $A, B \\subseteq \\mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \\subseteq \\mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\\{0, 1, \\ldots, k\\}$ which have $0$ as an element, the full interval $\\{0, 1, \\ldots,k\\}$ has the most divisors. To generalize to sets which do not have $0$ as an element, we prove a correspondence between additive divisors and lunar multiplication, introduced by Appelgate, LeBrun and Sloane (2011) in their study of a kind of min/max arithmetic. The number of binary lunar divisor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05220","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/1908.05220/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":"1908.05220","created_at":"2026-07-05T09:09:47.743627+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.05220v1","created_at":"2026-07-05T09:09:47.743627+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05220","created_at":"2026-07-05T09:09:47.743627+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZTOFIJY4B3AK","created_at":"2026-07-05T09:09:47.743627+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZTOFIJY4B3AKVSRB","created_at":"2026-07-05T09:09:47.743627+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZTOFIJY4","created_at":"2026-07-05T09:09:47.743627+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/ZTOFIJY4B3AKVSRBVQ4FFHA3LR","json":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR.json","graph_json":"https://pith.science/api/pith-number/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/graph.json","events_json":"https://pith.science/api/pith-number/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/events.json","paper":"https://pith.science/paper/ZTOFIJY4"},"agent_actions":{"view_html":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR","download_json":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR.json","view_paper":"https://pith.science/paper/ZTOFIJY4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.05220&json=true","fetch_graph":"https://pith.science/api/pith-number/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/graph.json","fetch_events":"https://pith.science/api/pith-number/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/action/storage_attestation","attest_author":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/action/author_attestation","sign_citation":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/action/citation_signature","submit_replication":"https://pith.science/pith/ZTOFIJY4B3AKVSRBVQ4FFHA3LR/action/replication_record"}},"created_at":"2026-07-05T09:09:47.743627+00:00","updated_at":"2026-07-05T09:09:47.743627+00:00"}