{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:UIICJDZNG4TVTXUDHH5OSHYFFD","short_pith_number":"pith:UIICJDZN","canonical_record":{"source":{"id":"1910.12664","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-10-28T13:31:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e1b3a619472679a0950a8f68166045683551febf601943f2ff814768e4c3242d","abstract_canon_sha256":"f4b812ff38497dd54088209613a097a40530ac2923346f47a6540b86aecf3e12"},"schema_version":"1.0"},"canonical_sha256":"a210248f2d372759de8339fae91f0528e764231019698f30aa7ca6f05e0bda08","source":{"kind":"arxiv","id":"1910.12664","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.12664","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1910.12664v3","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.12664","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"UIICJDZNG4TV","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_16","alias_value":"UIICJDZNG4TVTXUD","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_8","alias_value":"UIICJDZN","created_at":"2026-07-05T02:04:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:UIICJDZNG4TVTXUDHH5OSHYFFD","target":"record","payload":{"canonical_record":{"source":{"id":"1910.12664","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-10-28T13:31:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e1b3a619472679a0950a8f68166045683551febf601943f2ff814768e4c3242d","abstract_canon_sha256":"f4b812ff38497dd54088209613a097a40530ac2923346f47a6540b86aecf3e12"},"schema_version":"1.0"},"canonical_sha256":"a210248f2d372759de8339fae91f0528e764231019698f30aa7ca6f05e0bda08","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:04:26.144861Z","signature_b64":"QeSLbsuRgUaqLrx4Vlscc0o98u+hWR0RymDxXh6AQH93bsleRwKPXkQ+EYJF8N8caVWuO5IZwmZ49JFCcn5xBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a210248f2d372759de8339fae91f0528e764231019698f30aa7ca6f05e0bda08","last_reissued_at":"2026-07-05T02:04:26.144435Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:04:26.144435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1910.12664","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CTU7pWmsyFjuhKqwEk14+vy30QF7Q12Y7mpkS2xlK4LMHzjejebovGzbs4PYBpM47gHNBNqhS9eyQfpFghFMBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T10:03:24.762001Z"},"content_sha256":"d8e7ae1ea678271cd3031e7c55354e7bf8631f3cfd17972b4c847f60143b4994","schema_version":"1.0","event_id":"sha256:d8e7ae1ea678271cd3031e7c55354e7bf8631f3cfd17972b4c847f60143b4994"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:UIICJDZNG4TVTXUDHH5OSHYFFD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Waring's problem over finite fields through generalized Paley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","submitted_at":"2019-10-28T13:31:49Z","abstract_excerpt":"We show that the Waring's number over a finite field $\\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $\\Gamma(k,q)=Cay(\\mathbb{F}_{q},R_k)$ with $R_k=\\{x^k : x\\in \\mathbb{F}_q^*\\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $\\Gamma(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.12664","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1910.12664/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5lrZFcBlUu+P0AoKxpMT8NSy9nKQoxP8vYpcAKAeYuDZi5zIw8RQfNOvkaqeH9t6hFt21ygGNIFfXNmm43i0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T10:03:24.762379Z"},"content_sha256":"6f1694b4ed943c572e8b08d80514d7c545bd067b0fa37c0f8d865a1ae557df62","schema_version":"1.0","event_id":"sha256:6f1694b4ed943c572e8b08d80514d7c545bd067b0fa37c0f8d865a1ae557df62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/bundle.json","state_url":"https://pith.science/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/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":"2026-08-23T10:03:24Z","links":{"resolver":"https://pith.science/pith/UIICJDZNG4TVTXUDHH5OSHYFFD","bundle":"https://pith.science/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/bundle.json","state":"https://pith.science/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UIICJDZNG4TVTXUDHH5OSHYFFD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:UIICJDZNG4TVTXUDHH5OSHYFFD","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":"f4b812ff38497dd54088209613a097a40530ac2923346f47a6540b86aecf3e12","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-10-28T13:31:49Z","title_canon_sha256":"e1b3a619472679a0950a8f68166045683551febf601943f2ff814768e4c3242d"},"schema_version":"1.0","source":{"id":"1910.12664","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.12664","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1910.12664v3","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.12664","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"UIICJDZNG4TV","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_16","alias_value":"UIICJDZNG4TVTXUD","created_at":"2026-07-05T02:04:26Z"},{"alias_kind":"pith_short_8","alias_value":"UIICJDZN","created_at":"2026-07-05T02:04:26Z"}],"graph_snapshots":[{"event_id":"sha256:6f1694b4ed943c572e8b08d80514d7c545bd067b0fa37c0f8d865a1ae557df62","target":"graph","created_at":"2026-07-05T02:04:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1910.12664/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the Waring's number over a finite field $\\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $\\Gamma(k,q)=Cay(\\mathbb{F}_{q},R_k)$ with $R_k=\\{x^k : x\\in \\mathbb{F}_q^*\\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $\\Gamma(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-10-28T13:31:49Z","title":"The Waring's problem over finite fields through generalized Paley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.12664","kind":"arxiv","version":3},"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:d8e7ae1ea678271cd3031e7c55354e7bf8631f3cfd17972b4c847f60143b4994","target":"record","created_at":"2026-07-05T02:04:26Z","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":"f4b812ff38497dd54088209613a097a40530ac2923346f47a6540b86aecf3e12","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-10-28T13:31:49Z","title_canon_sha256":"e1b3a619472679a0950a8f68166045683551febf601943f2ff814768e4c3242d"},"schema_version":"1.0","source":{"id":"1910.12664","kind":"arxiv","version":3}},"canonical_sha256":"a210248f2d372759de8339fae91f0528e764231019698f30aa7ca6f05e0bda08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a210248f2d372759de8339fae91f0528e764231019698f30aa7ca6f05e0bda08","first_computed_at":"2026-07-05T02:04:26.144435Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:04:26.144435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QeSLbsuRgUaqLrx4Vlscc0o98u+hWR0RymDxXh6AQH93bsleRwKPXkQ+EYJF8N8caVWuO5IZwmZ49JFCcn5xBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:04:26.144861Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.12664","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d8e7ae1ea678271cd3031e7c55354e7bf8631f3cfd17972b4c847f60143b4994","sha256:6f1694b4ed943c572e8b08d80514d7c545bd067b0fa37c0f8d865a1ae557df62"],"state_sha256":"c91612149f33ddedc739a37e667635919570a9db15ba86da3b681588f1801759"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ufyX+ZYvEwZeGsk7SiHkgFIcMoJioUjASlSHFPriE72fai7RyroVt4Bm4yFD2CbscJYxXGXc/IS19FoqSS5kCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T10:03:24.765007Z","bundle_sha256":"1344229b3681b6ab04d948b3f08dc89886bfae816c741d34c248ab214a6d2b44"}}