{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6SYXBVMRC2PCT4VYLVSL5FWBOQ","short_pith_number":"pith:6SYXBVMR","canonical_record":{"source":{"id":"1109.2860","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-13T17:39:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"59cdb0c7904338a64c7df6a9de8e3280de09129769860251e6ca990471149c07","abstract_canon_sha256":"348055e640d8025c102969c7a1bb3458687443fa4d5c4d04293063a60635304d"},"schema_version":"1.0"},"canonical_sha256":"f4b170d591169e29f2b85d64be96c174182a8704096df1e701849433c3d0bf99","source":{"kind":"arxiv","id":"1109.2860","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2860","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2860v1","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2860","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"pith_short_12","alias_value":"6SYXBVMRC2PC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6SYXBVMRC2PCT4VY","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6SYXBVMR","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6SYXBVMRC2PCT4VYLVSL5FWBOQ","target":"record","payload":{"canonical_record":{"source":{"id":"1109.2860","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-13T17:39:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"59cdb0c7904338a64c7df6a9de8e3280de09129769860251e6ca990471149c07","abstract_canon_sha256":"348055e640d8025c102969c7a1bb3458687443fa4d5c4d04293063a60635304d"},"schema_version":"1.0"},"canonical_sha256":"f4b170d591169e29f2b85d64be96c174182a8704096df1e701849433c3d0bf99","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:07.545757Z","signature_b64":"ghRl7hSSKwYIkRzFtRF34CT90Cl7v1pDqjZuCBXqSjfYXVI0kVqUOxlj6mEikHLsSVV2c2walVKiwSB4lOEvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4b170d591169e29f2b85d64be96c174182a8704096df1e701849433c3d0bf99","last_reissued_at":"2026-05-18T04:13:07.545155Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:07.545155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.2860","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:13:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yHrBc0hxYR7UHtiBo+hibW5X1d5hoG8/MaZLoLEMjGHQklx8g353qRJtdbJ8ixZJ0YiL8QAHq3WGAVUw9CFkBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:33:51.761656Z"},"content_sha256":"40bf18d3bc32465acf91491d85c781e22901bc009d413f40ded648aa30145843","schema_version":"1.0","event_id":"sha256:40bf18d3bc32465acf91491d85c781e22901bc009d413f40ded648aa30145843"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6SYXBVMRC2PCT4VYLVSL5FWBOQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Calculation of norms of some secial elements of cyclotomic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Alexandre Aksenov","submitted_at":"2011-09-13T17:39:08Z","abstract_excerpt":"In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2860","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:13:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EB2GzHGXSgb2r5Lg3bT2Tgc/ZJh5XVg+UN0bIdfndnvKr24MT2fEHQmVPcmvlNbnQ0+RqzgfdqRDMZ4fNzDhAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:33:51.762327Z"},"content_sha256":"518a3f1f60ad0a2766908695e4ae0df2f7ff7eec5fefa46db0ed7c4859bcabde","schema_version":"1.0","event_id":"sha256:518a3f1f60ad0a2766908695e4ae0df2f7ff7eec5fefa46db0ed7c4859bcabde"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/bundle.json","state_url":"https://pith.science/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/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":"2026-06-10T23:33:51Z","links":{"resolver":"https://pith.science/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ","bundle":"https://pith.science/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/bundle.json","state":"https://pith.science/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6SYXBVMRC2PCT4VYLVSL5FWBOQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6SYXBVMRC2PCT4VYLVSL5FWBOQ","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":"348055e640d8025c102969c7a1bb3458687443fa4d5c4d04293063a60635304d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-13T17:39:08Z","title_canon_sha256":"59cdb0c7904338a64c7df6a9de8e3280de09129769860251e6ca990471149c07"},"schema_version":"1.0","source":{"id":"1109.2860","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2860","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2860v1","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2860","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"pith_short_12","alias_value":"6SYXBVMRC2PC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6SYXBVMRC2PCT4VY","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6SYXBVMR","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:518a3f1f60ad0a2766908695e4ae0df2f7ff7eec5fefa46db0ed7c4859bcabde","target":"graph","created_at":"2026-05-18T04:13:07Z","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":"In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.","authors_text":"Alexandre Aksenov","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-13T17:39:08Z","title":"Calculation of norms of some secial elements of cyclotomic fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2860","kind":"arxiv","version":1},"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":"sha256:40bf18d3bc32465acf91491d85c781e22901bc009d413f40ded648aa30145843","target":"record","created_at":"2026-05-18T04:13:07Z","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":"348055e640d8025c102969c7a1bb3458687443fa4d5c4d04293063a60635304d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-13T17:39:08Z","title_canon_sha256":"59cdb0c7904338a64c7df6a9de8e3280de09129769860251e6ca990471149c07"},"schema_version":"1.0","source":{"id":"1109.2860","kind":"arxiv","version":1}},"canonical_sha256":"f4b170d591169e29f2b85d64be96c174182a8704096df1e701849433c3d0bf99","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4b170d591169e29f2b85d64be96c174182a8704096df1e701849433c3d0bf99","first_computed_at":"2026-05-18T04:13:07.545155Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:13:07.545155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ghRl7hSSKwYIkRzFtRF34CT90Cl7v1pDqjZuCBXqSjfYXVI0kVqUOxlj6mEikHLsSVV2c2walVKiwSB4lOEvDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:13:07.545757Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2860","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40bf18d3bc32465acf91491d85c781e22901bc009d413f40ded648aa30145843","sha256:518a3f1f60ad0a2766908695e4ae0df2f7ff7eec5fefa46db0ed7c4859bcabde"],"state_sha256":"1a75925d89b6fd1140e7945ef50f06ae3285aae38919e01e8abccd395ba14a6b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bzSHaITB7R+azP1rgLMa74GxTtUiEPPnbk7MsWXiNL9/QWWtZGK/4EXbVJY9qB4PT6dNBYUf3SP1cy8d3L2gCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:33:51.765950Z","bundle_sha256":"e395fc19029dc01a6b9623f42847812f284f5c3745a38abf866e294c12b02bea"}}