{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:6CMW6N2IUJ33IBIO7L3C5A5LTR","short_pith_number":"pith:6CMW6N2I","canonical_record":{"source":{"id":"1410.7073","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-26T20:10:59Z","cross_cats_sorted":[],"title_canon_sha256":"5212a810df38bd4fd1a87197dd3b0566bb0257f79298f5adcad3715b73552976","abstract_canon_sha256":"a84d5b98b5d9e288c7f46a289c5fd56a3662d497887acc24c769d08891d56bd6"},"schema_version":"1.0"},"canonical_sha256":"f0996f3748a277b4050efaf62e83ab9c7afb430e8aeb8addae63d3dcb9188475","source":{"kind":"arxiv","id":"1410.7073","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7073","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7073v3","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7073","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"6CMW6N2IUJ33","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6CMW6N2IUJ33IBIO","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6CMW6N2I","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:6CMW6N2IUJ33IBIO7L3C5A5LTR","target":"record","payload":{"canonical_record":{"source":{"id":"1410.7073","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-26T20:10:59Z","cross_cats_sorted":[],"title_canon_sha256":"5212a810df38bd4fd1a87197dd3b0566bb0257f79298f5adcad3715b73552976","abstract_canon_sha256":"a84d5b98b5d9e288c7f46a289c5fd56a3662d497887acc24c769d08891d56bd6"},"schema_version":"1.0"},"canonical_sha256":"f0996f3748a277b4050efaf62e83ab9c7afb430e8aeb8addae63d3dcb9188475","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:21.647745Z","signature_b64":"XqzzMBeylDRt08TurLF/aQ5afTVgK/9Nn7ArNCq3JzUJyuNePrZ03LEdtLvIEUUQ08KRQKEbMQWdTOc25AdkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0996f3748a277b4050efaf62e83ab9c7afb430e8aeb8addae63d3dcb9188475","last_reissued_at":"2026-05-18T01:22:21.647074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:21.647074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.7073","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":"2026-05-18T01:22:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zGMSnaVmZoHW9oCBm4X9VdRyalWBwpyeFzwd3tqKFLxwT4lYOv2+pMqdxJG5X00Jz+W6Oh+bx1B3aDh1lkoUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T11:08:47.175420Z"},"content_sha256":"c547d3ad95b6c8843376343d0919fa552da34500f4d22470f1d0e6a7eeca5f56","schema_version":"1.0","event_id":"sha256:c547d3ad95b6c8843376343d0919fa552da34500f4d22470f1d0e6a7eeca5f56"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:6CMW6N2IUJ33IBIO7L3C5A5LTR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Terence Tao","submitted_at":"2014-10-26T20:10:59Z","abstract_excerpt":"For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\\eps)$ for every fixed $\\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost all primes $p$ but remains open in general. In this paper we show that Vinogradov's conjecture also follows from the Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential \"non-multiplicative\" route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7073","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":""},"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":"2026-05-18T01:22:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9kxxrWzCrK/+j5eKxFl4R0k1MXIqmqtjuEip8J9Hrdtqf+q37l9oHuVUOEcZ09sUlSimuXP24nrGNHKWBVfQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T11:08:47.175768Z"},"content_sha256":"62b0de9ce4f1bff9c8258dfd3275ab8bfc4c4a284769e04a5b92e41db899fdd2","schema_version":"1.0","event_id":"sha256:62b0de9ce4f1bff9c8258dfd3275ab8bfc4c4a284769e04a5b92e41db899fdd2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/bundle.json","state_url":"https://pith.science/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/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":"2026-06-05T11:08:47Z","links":{"resolver":"https://pith.science/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR","bundle":"https://pith.science/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/bundle.json","state":"https://pith.science/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6CMW6N2IUJ33IBIO7L3C5A5LTR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6CMW6N2IUJ33IBIO7L3C5A5LTR","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":"a84d5b98b5d9e288c7f46a289c5fd56a3662d497887acc24c769d08891d56bd6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-26T20:10:59Z","title_canon_sha256":"5212a810df38bd4fd1a87197dd3b0566bb0257f79298f5adcad3715b73552976"},"schema_version":"1.0","source":{"id":"1410.7073","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7073","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7073v3","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7073","created_at":"2026-05-18T01:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"6CMW6N2IUJ33","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6CMW6N2IUJ33IBIO","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6CMW6N2I","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:62b0de9ce4f1bff9c8258dfd3275ab8bfc4c4a284769e04a5b92e41db899fdd2","target":"graph","created_at":"2026-05-18T01:22:21Z","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 each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\\eps)$ for every fixed $\\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost all primes $p$ but remains open in general. In this paper we show that Vinogradov's conjecture also follows from the Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential \"non-multiplicative\" route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds ","authors_text":"Terence Tao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-26T20:10:59Z","title":"The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7073","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:c547d3ad95b6c8843376343d0919fa552da34500f4d22470f1d0e6a7eeca5f56","target":"record","created_at":"2026-05-18T01:22:21Z","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":"a84d5b98b5d9e288c7f46a289c5fd56a3662d497887acc24c769d08891d56bd6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-26T20:10:59Z","title_canon_sha256":"5212a810df38bd4fd1a87197dd3b0566bb0257f79298f5adcad3715b73552976"},"schema_version":"1.0","source":{"id":"1410.7073","kind":"arxiv","version":3}},"canonical_sha256":"f0996f3748a277b4050efaf62e83ab9c7afb430e8aeb8addae63d3dcb9188475","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0996f3748a277b4050efaf62e83ab9c7afb430e8aeb8addae63d3dcb9188475","first_computed_at":"2026-05-18T01:22:21.647074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:21.647074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XqzzMBeylDRt08TurLF/aQ5afTVgK/9Nn7ArNCq3JzUJyuNePrZ03LEdtLvIEUUQ08KRQKEbMQWdTOc25AdkAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:21.647745Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.7073","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c547d3ad95b6c8843376343d0919fa552da34500f4d22470f1d0e6a7eeca5f56","sha256:62b0de9ce4f1bff9c8258dfd3275ab8bfc4c4a284769e04a5b92e41db899fdd2"],"state_sha256":"0d74eff4ae2bf4bc983313a76c784d6102de166f37d30a6fa5616f6759639a22"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mWD2TYvJuBsJdUox5YaktJyL22d/RyE6Pjm1XoHLTnXSPMBuOgkHzq7OBZ9GWOdS4KAylxpidpviWHc3Mv0HAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T11:08:47.178112Z","bundle_sha256":"71155941953db1cf3950f7f5bd30750b32c83f772494a440dfe9a5ab31e8de51"}}