{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:S7UTMHXBGT5VQTVZSZ4TVOS2GH","short_pith_number":"pith:S7UTMHXB","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"},"canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","source":{"kind":"arxiv","id":"1003.0021","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0021","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0021v2","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0021","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"S7UTMHXBGT5V","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"S7UTMHXBGT5VQTVZ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"S7UTMHXB","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:S7UTMHXBGT5VQTVZSZ4TVOS2GH","target":"record","payload":{"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"},"canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","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"},"source_kind":"arxiv","source_id":"1003.0021","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+bWZzz2v3Ns0PQl9rWf/i/jxsYqg221cqXnx+QR7uJ2W3Olx8+4qlS0wqlhCwxdWK9tZedZxb+vt6QKwGgkGAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T15:25:24.803079Z"},"content_sha256":"810fb93314c026b5fc2401885617ef39edf5a196b3e922cbbc1a02cf830e24ab","schema_version":"1.0","event_id":"sha256:810fb93314c026b5fc2401885617ef39edf5a196b3e922cbbc1a02cf830e24ab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:S7UTMHXBGT5VQTVZSZ4TVOS2GH","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a6pT5cld4LJ0BGU1MZhezeBEnoI+yYVmxgN0XbK7sxJPG/Xxrk7gNnxEItqzHSUxZGbROCb5STuqWWpumw6NCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T15:25:24.803592Z"},"content_sha256":"3ada9aa5b36faec93603cc064d07b8476c1997cf0c997402b1890bcc00db864d","schema_version":"1.0","event_id":"sha256:3ada9aa5b36faec93603cc064d07b8476c1997cf0c997402b1890bcc00db864d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/bundle.json","state_url":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-22T15:25:24Z","links":{"resolver":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH","bundle":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/bundle.json","state":"https://pith.science/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S7UTMHXBGT5VQTVZSZ4TVOS2GH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:S7UTMHXBGT5VQTVZSZ4TVOS2GH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9896697a45319c7f0b4d735667e7a8c2e96401ef446616e2c269300f9c752204","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-26T21:49:04Z","title_canon_sha256":"70bb5101b542a0bcf09200f0371220412111e0bc52238a3267465df12977d691"},"schema_version":"1.0","source":{"id":"1003.0021","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0021","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0021v2","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0021","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"S7UTMHXBGT5V","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"S7UTMHXBGT5VQTVZ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"S7UTMHXB","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:3ada9aa5b36faec93603cc064d07b8476c1997cf0c997402b1890bcc00db864d","target":"graph","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"James Stankewicz, Jeffrey Shallit","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-26T21:49:04Z","title":"Unbounded discrepancy in Frobenius numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0021","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:810fb93314c026b5fc2401885617ef39edf5a196b3e922cbbc1a02cf830e24ab","target":"record","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9896697a45319c7f0b4d735667e7a8c2e96401ef446616e2c269300f9c752204","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-26T21:49:04Z","title_canon_sha256":"70bb5101b542a0bcf09200f0371220412111e0bc52238a3267465df12977d691"},"schema_version":"1.0","source":{"id":"1003.0021","kind":"arxiv","version":2}},"canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97e9361ee134fb584eb996793aba5a31c5b6faa294424c559fa7a43d8726e3b8","first_computed_at":"2026-05-18T04:41:16.270901Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:16.270901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cJmPBQx65kjiJn8BGdCXLhLsZ08D9qg6iesnuXe5VW2gh+CstFtP/IEYIc8HxgTB0BV2NzkR1OfqrC1ydlczDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:16.271747Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.0021","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:810fb93314c026b5fc2401885617ef39edf5a196b3e922cbbc1a02cf830e24ab","sha256:3ada9aa5b36faec93603cc064d07b8476c1997cf0c997402b1890bcc00db864d"],"state_sha256":"3bc4e91ede787d09a9f3bd94fce236ab264a8fae8c680b2d0b391950cc504508"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mVwNxOsI/MWMdCuN6tRjF2f8UNPZa5zhzbZW8F1lMxAfpqMPtTdUFQeBtl4QYGzUg8qjlkq/i2cvtehkIqzEDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T15:25:24.806102Z","bundle_sha256":"936ec45d6c4003f4fd7b4c9404790f78cc64e8704bb98ae5ed2ff342202bf5ab"}}