{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:TTIZRIR5FN3GLMSMO7ZMDAO33L","short_pith_number":"pith:TTIZRIR5","canonical_record":{"source":{"id":"1310.1772","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-07T13:17:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"69a4e4c85741fc57d9dcc308dd201c38a61393e4890ace02008663d56927b7bb","abstract_canon_sha256":"80a3dfa6485e938e6c76d58cfe3a0df250a0504b9f47c165dbbbee59677f830e"},"schema_version":"1.0"},"canonical_sha256":"9cd198a23d2b7665b24c77f2c181dbdac5478398fa7affb5e74a839db2561114","source":{"kind":"arxiv","id":"1310.1772","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.1772","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"arxiv_version","alias_value":"1310.1772v1","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1772","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"pith_short_12","alias_value":"TTIZRIR5FN3G","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"TTIZRIR5FN3GLMSM","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"TTIZRIR5","created_at":"2026-05-18T12:28:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:TTIZRIR5FN3GLMSMO7ZMDAO33L","target":"record","payload":{"canonical_record":{"source":{"id":"1310.1772","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-07T13:17:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"69a4e4c85741fc57d9dcc308dd201c38a61393e4890ace02008663d56927b7bb","abstract_canon_sha256":"80a3dfa6485e938e6c76d58cfe3a0df250a0504b9f47c165dbbbee59677f830e"},"schema_version":"1.0"},"canonical_sha256":"9cd198a23d2b7665b24c77f2c181dbdac5478398fa7affb5e74a839db2561114","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:42.681646Z","signature_b64":"tcV27MjbmBJNSfv3pIwZzfKRbH0UkeyIsSnBnq9/Vo8RfGYIAzFSnz+mxt+zUhcbaa0U+Y+q1uZfL2CGb5/8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cd198a23d2b7665b24c77f2c181dbdac5478398fa7affb5e74a839db2561114","last_reissued_at":"2026-05-18T01:19:42.681128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:42.681128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.1772","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":"2026-05-18T01:19:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MuUsz1q1xzTu+mkdOQTCYmJG5f5GbYfUvsWddYYeUbe9hpXqipGAH2WvQm9hXWgB1HgtlqfJNDPS8mXySsiBDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:55:11.495107Z"},"content_sha256":"257e493b899b585466c5fec9c61f9913c43bd1f4bf9171648566926c493f8db2","schema_version":"1.0","event_id":"sha256:257e493b899b585466c5fec9c61f9913c43bd1f4bf9171648566926c493f8db2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:TTIZRIR5FN3GLMSMO7ZMDAO33L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rational points on some Fermat curves and surfaces over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Jose Felipe Voloch, Michael E. Zieve","submitted_at":"2013-10-07T13:17:34Z","abstract_excerpt":"We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also describe the F_{q^2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates of any such points is a square."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1772","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":"2026-05-18T01:19:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yfSeRQtwKSbG7PCrsGOtHnjOYJsqEbGvJMiOwH43aXguSJRGtXAfvs2ULKLj83EElBo2leEIpkyDubpUoxlbDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:55:11.495812Z"},"content_sha256":"770de74d6f24b9f809aea9e01c7c39f434e45622e7a27355ecf812af25d36c39","schema_version":"1.0","event_id":"sha256:770de74d6f24b9f809aea9e01c7c39f434e45622e7a27355ecf812af25d36c39"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/bundle.json","state_url":"https://pith.science/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/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":"2026-06-04T22:55:11Z","links":{"resolver":"https://pith.science/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L","bundle":"https://pith.science/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/bundle.json","state":"https://pith.science/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TTIZRIR5FN3GLMSMO7ZMDAO33L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:TTIZRIR5FN3GLMSMO7ZMDAO33L","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":"80a3dfa6485e938e6c76d58cfe3a0df250a0504b9f47c165dbbbee59677f830e","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-07T13:17:34Z","title_canon_sha256":"69a4e4c85741fc57d9dcc308dd201c38a61393e4890ace02008663d56927b7bb"},"schema_version":"1.0","source":{"id":"1310.1772","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.1772","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"arxiv_version","alias_value":"1310.1772v1","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1772","created_at":"2026-05-18T01:19:42Z"},{"alias_kind":"pith_short_12","alias_value":"TTIZRIR5FN3G","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"TTIZRIR5FN3GLMSM","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"TTIZRIR5","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:770de74d6f24b9f809aea9e01c7c39f434e45622e7a27355ecf812af25d36c39","target":"graph","created_at":"2026-05-18T01:19:42Z","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":"We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also describe the F_{q^2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates of any such points is a square.","authors_text":"Jose Felipe Voloch, Michael E. Zieve","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-07T13:17:34Z","title":"Rational points on some Fermat curves and surfaces over finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1772","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":"sha256:257e493b899b585466c5fec9c61f9913c43bd1f4bf9171648566926c493f8db2","target":"record","created_at":"2026-05-18T01:19:42Z","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":"80a3dfa6485e938e6c76d58cfe3a0df250a0504b9f47c165dbbbee59677f830e","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-07T13:17:34Z","title_canon_sha256":"69a4e4c85741fc57d9dcc308dd201c38a61393e4890ace02008663d56927b7bb"},"schema_version":"1.0","source":{"id":"1310.1772","kind":"arxiv","version":1}},"canonical_sha256":"9cd198a23d2b7665b24c77f2c181dbdac5478398fa7affb5e74a839db2561114","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9cd198a23d2b7665b24c77f2c181dbdac5478398fa7affb5e74a839db2561114","first_computed_at":"2026-05-18T01:19:42.681128Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:42.681128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tcV27MjbmBJNSfv3pIwZzfKRbH0UkeyIsSnBnq9/Vo8RfGYIAzFSnz+mxt+zUhcbaa0U+Y+q1uZfL2CGb5/8DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:42.681646Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.1772","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:257e493b899b585466c5fec9c61f9913c43bd1f4bf9171648566926c493f8db2","sha256:770de74d6f24b9f809aea9e01c7c39f434e45622e7a27355ecf812af25d36c39"],"state_sha256":"ac43e34797fcc93a393b8064cea649d01e5b593758103fff5f9abe6e49376fb8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dtoz/34aYSLHzGmtG5ouZxzzIS68hKEwYKAPAp8JHHZLxtfEOrsm34lLpX8LmvfjV/NyxpycnrJ/qiyCTm2pBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T22:55:11.499918Z","bundle_sha256":"a86c3bbf03f4dd25c9b6ef504a0d98cb453c26c63d4be9aad2cc0de2c436ec1d"}}