{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:NVS5TSUK4MBZJQPXTHQDODO45K","short_pith_number":"pith:NVS5TSUK","canonical_record":{"source":{"id":"1903.07314","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-18T09:13:47Z","cross_cats_sorted":[],"title_canon_sha256":"ab37c3a673283e7abd633eaa68938f8b5027fa5a3e5734edb0d7ee2015829a82","abstract_canon_sha256":"add17071fe1078e6dfc8497fc8949a40f4373a72ffb9a575e56eedb71594656f"},"schema_version":"1.0"},"canonical_sha256":"6d65d9ca8ae30394c1f799e0370ddcea817a522f47569252e7778bc843ef1cae","source":{"kind":"arxiv","id":"1903.07314","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.07314","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.07314v1","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.07314","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"pith_short_12","alias_value":"NVS5TSUK4MBZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"NVS5TSUK4MBZJQPX","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"NVS5TSUK","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:NVS5TSUK4MBZJQPXTHQDODO45K","target":"record","payload":{"canonical_record":{"source":{"id":"1903.07314","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-18T09:13:47Z","cross_cats_sorted":[],"title_canon_sha256":"ab37c3a673283e7abd633eaa68938f8b5027fa5a3e5734edb0d7ee2015829a82","abstract_canon_sha256":"add17071fe1078e6dfc8497fc8949a40f4373a72ffb9a575e56eedb71594656f"},"schema_version":"1.0"},"canonical_sha256":"6d65d9ca8ae30394c1f799e0370ddcea817a522f47569252e7778bc843ef1cae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:01.768742Z","signature_b64":"lmEu1Tf+MNIcncQS75Qv8RiPKMCvN0BzZDqla7zp1xEdSgd2uuiA3l2Gg45pbjyKMIWJdJs/HInOCArPgp1kAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d65d9ca8ae30394c1f799e0370ddcea817a522f47569252e7778bc843ef1cae","last_reissued_at":"2026-05-17T23:51:01.768215Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:01.768215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.07314","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":"2026-05-17T23:51:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l0+/S6V+mZue/QnPwY/kRNF9zXADUnbfzipLAA1hm4yI3TDcaak84mWJiwgPMr1s+1VTZ46N+CBM6SJa6FGUBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T23:30:26.807234Z"},"content_sha256":"9b287df2d1a8ccecc75c25ad937418f61a954c512310875707c9acb5f7e56c62","schema_version":"1.0","event_id":"sha256:9b287df2d1a8ccecc75c25ad937418f61a954c512310875707c9acb5f7e56c62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:NVS5TSUK4MBZJQPXTHQDODO45K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Upper Bounds for Cyclotomic Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bernhard Schmidt, Ka Hin Leung, Tai Do Duc","submitted_at":"2019-03-18T09:13:47Z","abstract_excerpt":"Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\\mathbb{F}_q$. A general result of our study is that $(a,b)\\leq 3$ for all $a,b \\in \\mathbb{Z}$ if $p> (\\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\\neq b$ and $a,b \\in \\{1,\\dots,e-1\\}$. The main idea we use is to transform equations over $\\mathbb{F}_q$ into equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07314","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":"2026-05-17T23:51:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KhcBF30CWR0p9j+j/rIWNFTzAHIiDyEPI3jkLsol63qkEnJUbA3CfgkCxxlW29i+COK042FE+FMxcXBGKT6FCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T23:30:26.807588Z"},"content_sha256":"c5c8bd6dc5e1e6e642f5d368d551bf4fb39305868918565fd351e5355fc01237","schema_version":"1.0","event_id":"sha256:c5c8bd6dc5e1e6e642f5d368d551bf4fb39305868918565fd351e5355fc01237"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NVS5TSUK4MBZJQPXTHQDODO45K/bundle.json","state_url":"https://pith.science/pith/NVS5TSUK4MBZJQPXTHQDODO45K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NVS5TSUK4MBZJQPXTHQDODO45K/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":"2026-06-21T23:30:26Z","links":{"resolver":"https://pith.science/pith/NVS5TSUK4MBZJQPXTHQDODO45K","bundle":"https://pith.science/pith/NVS5TSUK4MBZJQPXTHQDODO45K/bundle.json","state":"https://pith.science/pith/NVS5TSUK4MBZJQPXTHQDODO45K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NVS5TSUK4MBZJQPXTHQDODO45K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:NVS5TSUK4MBZJQPXTHQDODO45K","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":"add17071fe1078e6dfc8497fc8949a40f4373a72ffb9a575e56eedb71594656f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-18T09:13:47Z","title_canon_sha256":"ab37c3a673283e7abd633eaa68938f8b5027fa5a3e5734edb0d7ee2015829a82"},"schema_version":"1.0","source":{"id":"1903.07314","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.07314","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.07314v1","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.07314","created_at":"2026-05-17T23:51:01Z"},{"alias_kind":"pith_short_12","alias_value":"NVS5TSUK4MBZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"NVS5TSUK4MBZJQPX","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"NVS5TSUK","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:c5c8bd6dc5e1e6e642f5d368d551bf4fb39305868918565fd351e5355fc01237","target":"graph","created_at":"2026-05-17T23:51:01Z","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 $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\\mathbb{F}_q$. A general result of our study is that $(a,b)\\leq 3$ for all $a,b \\in \\mathbb{Z}$ if $p> (\\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\\neq b$ and $a,b \\in \\{1,\\dots,e-1\\}$. The main idea we use is to transform equations over $\\mathbb{F}_q$ into equation","authors_text":"Bernhard Schmidt, Ka Hin Leung, Tai Do Duc","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-18T09:13:47Z","title":"Upper Bounds for Cyclotomic Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07314","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":"sha256:9b287df2d1a8ccecc75c25ad937418f61a954c512310875707c9acb5f7e56c62","target":"record","created_at":"2026-05-17T23:51:01Z","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":"add17071fe1078e6dfc8497fc8949a40f4373a72ffb9a575e56eedb71594656f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-18T09:13:47Z","title_canon_sha256":"ab37c3a673283e7abd633eaa68938f8b5027fa5a3e5734edb0d7ee2015829a82"},"schema_version":"1.0","source":{"id":"1903.07314","kind":"arxiv","version":1}},"canonical_sha256":"6d65d9ca8ae30394c1f799e0370ddcea817a522f47569252e7778bc843ef1cae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d65d9ca8ae30394c1f799e0370ddcea817a522f47569252e7778bc843ef1cae","first_computed_at":"2026-05-17T23:51:01.768215Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:01.768215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lmEu1Tf+MNIcncQS75Qv8RiPKMCvN0BzZDqla7zp1xEdSgd2uuiA3l2Gg45pbjyKMIWJdJs/HInOCArPgp1kAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:01.768742Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.07314","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b287df2d1a8ccecc75c25ad937418f61a954c512310875707c9acb5f7e56c62","sha256:c5c8bd6dc5e1e6e642f5d368d551bf4fb39305868918565fd351e5355fc01237"],"state_sha256":"cd40110c613324681194b9e9226507d0923d13258f41620f2277e0743648cf13"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJjVu7sajbLNRxDxwt+/AOhu4VRpJrR1Tlb8W819xC/3kAa2gwbohAXV/SDgFUzabp1SCgamPLvRqmaiXI8bBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T23:30:26.809487Z","bundle_sha256":"2792cc1750c655565893f205473c7b1fbb8d790684d079f180b4e2415ff2a767"}}