{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:S7UTMHXBGT5VQTVZSZ4TVOS2GH","short_pith_number":"pith:S7UTMHXB","schema_version":"1.0","canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","source":{"kind":"arxiv","id":"1003.0021","version":2},"attestation_state":"computed","paper":{"title":"Unbounded discrepancy in Frobenius numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Stankewicz, Jeffrey Shallit","submitted_at":"2010-02-26T21:49:04Z","abstract_excerpt":"Let g_j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x_1, ... , x_n. We show that for any k > 0, and n = 5, the quantity g_0 - g_k is unbounded. Furthermore, we provide examples with g_0 > g_k for n >= 6 and g_0 > g_1 for n >= 4."},"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":"1003.0021","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-26T21:49:04Z","cross_cats_sorted":[],"title_canon_sha256":"70bb5101b542a0bcf09200f0371220412111e0bc52238a3267465df12977d691","abstract_canon_sha256":"9896697a45319c7f0b4d735667e7a8c2e96401ef446616e2c269300f9c752204"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:16.271747Z","signature_b64":"cJmPBQx65kjiJn8BGdCXLhLsZ08D9qg6iesnuXe5VW2gh+CstFtP/IEYIc8HxgTB0BV2NzkR1OfqrC1ydlczDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","last_reissued_at":"2026-05-18T04:41:16.270901Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:16.270901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unbounded discrepancy in Frobenius numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Stankewicz, Jeffrey Shallit","submitted_at":"2010-02-26T21:49:04Z","abstract_excerpt":"Let g_j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x_1, ... , x_n. We show that for any k > 0, and n = 5, the quantity g_0 - g_k is unbounded. Furthermore, we provide examples with g_0 > g_k for n >= 6 and g_0 > g_1 for n >= 4."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0021","kind":"arxiv","version":2},"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":"1003.0021","created_at":"2026-05-18T04:41:16.271214+00:00"},{"alias_kind":"arxiv_version","alias_value":"1003.0021v2","created_at":"2026-05-18T04:41:16.271214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0021","created_at":"2026-05-18T04:41:16.271214+00:00"},{"alias_kind":"pith_short_12","alias_value":"S7UTMHXBGT5V","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"S7UTMHXBGT5VQTVZ","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"S7UTMHXB","created_at":"2026-05-18T12:26:13.927090+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/S7UTMHXBGT5VQTVZSZ4TVOS2GH","json":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH.json","graph_json":"https://pith.science/api/pith-number/S7UTMHXBGT5VQTVZSZ4TVOS2GH/graph.json","events_json":"https://pith.science/api/pith-number/S7UTMHXBGT5VQTVZSZ4TVOS2GH/events.json","paper":"https://pith.science/paper/S7UTMHXB"},"agent_actions":{"view_html":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH","download_json":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH.json","view_paper":"https://pith.science/paper/S7UTMHXB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1003.0021&json=true","fetch_graph":"https://pith.science/api/pith-number/S7UTMHXBGT5VQTVZSZ4TVOS2GH/graph.json","fetch_events":"https://pith.science/api/pith-number/S7UTMHXBGT5VQTVZSZ4TVOS2GH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/action/storage_attestation","attest_author":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/action/author_attestation","sign_citation":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/action/citation_signature","submit_replication":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/action/replication_record"}},"created_at":"2026-05-18T04:41:16.271214+00:00","updated_at":"2026-05-18T04:41:16.271214+00:00"}