{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:NVHQSVVJJGFZYGCDB2EPQ4JF5T","short_pith_number":"pith:NVHQSVVJ","canonical_record":{"source":{"id":"1307.0453","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-01T17:55:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"362a472b7ad281c6f4f2338ede4df04b4dbdc28653299a1e7ed73603a90d1896","abstract_canon_sha256":"78282e581fccb2261712efaf6bcc26a4bf9a3f39f3bcfd2c3db3af99b1b7a3d6"},"schema_version":"1.0"},"canonical_sha256":"6d4f0956a9498b9c18430e88f87125ecdb9ff1dce8e25ba1e0091f1693c535d5","source":{"kind":"arxiv","id":"1307.0453","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0453","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0453v4","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0453","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"NVHQSVVJJGFZ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"NVHQSVVJJGFZYGCD","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"NVHQSVVJ","created_at":"2026-05-18T12:27:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:NVHQSVVJJGFZYGCDB2EPQ4JF5T","target":"record","payload":{"canonical_record":{"source":{"id":"1307.0453","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-01T17:55:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"362a472b7ad281c6f4f2338ede4df04b4dbdc28653299a1e7ed73603a90d1896","abstract_canon_sha256":"78282e581fccb2261712efaf6bcc26a4bf9a3f39f3bcfd2c3db3af99b1b7a3d6"},"schema_version":"1.0"},"canonical_sha256":"6d4f0956a9498b9c18430e88f87125ecdb9ff1dce8e25ba1e0091f1693c535d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:46.658566Z","signature_b64":"QQJ3an3S9GpnL5kdA5AxLkOWgG6lekojISmrU7u6JwHv7A1nwMbYvw6RA1ex7jd2RogZNR+h76xEOdqwVTCJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d4f0956a9498b9c18430e88f87125ecdb9ff1dce8e25ba1e0091f1693c535d5","last_reissued_at":"2026-05-18T02:42:46.657906Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:46.657906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.0453","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dutki0ycj8iB+ouaH5imWsBSo/MSFLSUXkrM3jN0uLrd8xAPhk9fdn/tG4YpP4jXJd8atOjIirYv/PVMFopVDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:10:35.283524Z"},"content_sha256":"96a010a15a8ce04013ae0c25367bfccbb88ae3bf8aea7417c34b696676ff8819","schema_version":"1.0","event_id":"sha256:96a010a15a8ce04013ae0c25367bfccbb88ae3bf8aea7417c34b696676ff8819"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:NVHQSVVJJGFZYGCDB2EPQ4JF5T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"2178 And All That","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"N. J. A. Sloane","submitted_at":"2013-07-01T17:55:29Z","abstract_excerpt":"For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0453","kind":"arxiv","version":4},"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":"2026-05-18T02:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eBRyk9Fg97SHzeR0o5UZnuEXj5s+eeEZ3lxh9v4XQwTRUcIetVdi+jrdn5fobzzXNx11tkdlmUndyeeB06/lBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:10:35.284137Z"},"content_sha256":"3b9523562eee6a643d5f8cafd8072ed10ce8a4bcb7645366213e78064778a88f","schema_version":"1.0","event_id":"sha256:3b9523562eee6a643d5f8cafd8072ed10ce8a4bcb7645366213e78064778a88f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/bundle.json","state_url":"https://pith.science/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/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":"2026-06-07T23:10:35Z","links":{"resolver":"https://pith.science/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T","bundle":"https://pith.science/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/bundle.json","state":"https://pith.science/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NVHQSVVJJGFZYGCDB2EPQ4JF5T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:NVHQSVVJJGFZYGCDB2EPQ4JF5T","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":"78282e581fccb2261712efaf6bcc26a4bf9a3f39f3bcfd2c3db3af99b1b7a3d6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-01T17:55:29Z","title_canon_sha256":"362a472b7ad281c6f4f2338ede4df04b4dbdc28653299a1e7ed73603a90d1896"},"schema_version":"1.0","source":{"id":"1307.0453","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0453","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0453v4","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0453","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"NVHQSVVJJGFZ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"NVHQSVVJJGFZYGCD","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"NVHQSVVJ","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:3b9523562eee6a643d5f8cafd8072ed10ce8a4bcb7645366213e78064778a88f","target":"graph","created_at":"2026-05-18T02:42:46Z","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":"For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples w","authors_text":"N. J. A. Sloane","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-01T17:55:29Z","title":"2178 And All That"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0453","kind":"arxiv","version":4},"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":"sha256:96a010a15a8ce04013ae0c25367bfccbb88ae3bf8aea7417c34b696676ff8819","target":"record","created_at":"2026-05-18T02:42:46Z","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":"78282e581fccb2261712efaf6bcc26a4bf9a3f39f3bcfd2c3db3af99b1b7a3d6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-01T17:55:29Z","title_canon_sha256":"362a472b7ad281c6f4f2338ede4df04b4dbdc28653299a1e7ed73603a90d1896"},"schema_version":"1.0","source":{"id":"1307.0453","kind":"arxiv","version":4}},"canonical_sha256":"6d4f0956a9498b9c18430e88f87125ecdb9ff1dce8e25ba1e0091f1693c535d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d4f0956a9498b9c18430e88f87125ecdb9ff1dce8e25ba1e0091f1693c535d5","first_computed_at":"2026-05-18T02:42:46.657906Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:46.657906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QQJ3an3S9GpnL5kdA5AxLkOWgG6lekojISmrU7u6JwHv7A1nwMbYvw6RA1ex7jd2RogZNR+h76xEOdqwVTCJCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:46.658566Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0453","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96a010a15a8ce04013ae0c25367bfccbb88ae3bf8aea7417c34b696676ff8819","sha256:3b9523562eee6a643d5f8cafd8072ed10ce8a4bcb7645366213e78064778a88f"],"state_sha256":"34b3bcc6d2be76bf27fb7837525200bcb7eca39f259c14ac391f70ce5bbdb726"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nTPtocXbR8n1Q97DayPJDj4cRLuieicVT2izlGfQWVW1378+/n34gkUwccTtE/AITsSH13eVn3D9k03Ez1gACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T23:10:35.287901Z","bundle_sha256":"54aae85a0edd114ea64845c5cb4a9361c3c84db57ba6466bfdad7f81f0569b27"}}