{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:OJ3UR442EZKJK4I67RT52MRPIX","short_pith_number":"pith:OJ3UR442","canonical_record":{"source":{"id":"1405.0290","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-21T19:36:18Z","cross_cats_sorted":[],"title_canon_sha256":"6ab402e820122973e0bed1aea058097a28ed0a1c51d69d509b5a962e726f3d01","abstract_canon_sha256":"c558a6f5263a69d0cba86942c89dd6ce9e6373b28dc2a867c0bd62aeefcacc35"},"schema_version":"1.0"},"canonical_sha256":"727748f39a265495711efc67dd322f45cf54c75f55568ea520409fa52a1f7800","source":{"kind":"arxiv","id":"1405.0290","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.0290","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"arxiv_version","alias_value":"1405.0290v5","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0290","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_12","alias_value":"OJ3UR442EZKJ","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_16","alias_value":"OJ3UR442EZKJK4I6","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_8","alias_value":"OJ3UR442","created_at":"2026-07-05T00:44:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:OJ3UR442EZKJK4I67RT52MRPIX","target":"record","payload":{"canonical_record":{"source":{"id":"1405.0290","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-21T19:36:18Z","cross_cats_sorted":[],"title_canon_sha256":"6ab402e820122973e0bed1aea058097a28ed0a1c51d69d509b5a962e726f3d01","abstract_canon_sha256":"c558a6f5263a69d0cba86942c89dd6ce9e6373b28dc2a867c0bd62aeefcacc35"},"schema_version":"1.0"},"canonical_sha256":"727748f39a265495711efc67dd322f45cf54c75f55568ea520409fa52a1f7800","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:44:23.343224Z","signature_b64":"wwQfVu1qdElb3sEzwpbpyURXbWuBKGbnSmGs8tdDBOsA3LpdryxX474bDlnKAUVZsfdFUse/sViJTW1PFfaBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"727748f39a265495711efc67dd322f45cf54c75f55568ea520409fa52a1f7800","last_reissued_at":"2026-07-05T00:44:23.342799Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:44:23.342799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.0290","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:44:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XhcjUaQ4sYZIhrHnl+OxvCpoK8DSfFBzpO+tJktzZJG8KIAeUZndT8FQ6WMS1PsawfMmfiU1wUjwlCAUq32tDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T22:32:32.789455Z"},"content_sha256":"e130d9762d80533246149a403513c936befa9145015b742a5a135544ff74ae5e","schema_version":"1.0","event_id":"sha256:e130d9762d80533246149a403513c936befa9145015b742a5a135544ff74ae5e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:OJ3UR442EZKJK4I67RT52MRPIX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New observations on primitive roots modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-04-21T19:36:18Z","abstract_excerpt":"We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\\sum_g(\\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\\ldots,p-1$, and $(\\frac{\\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0290","kind":"arxiv","version":5},"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/1405.0290/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":"2026-07-05T00:44:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UCuxOMvzM3M/ZNPG87bga0XmgxbrQeKYWLvax9Koiu/wIdG3e0MpvxisO2QBlb5XQccpTIZue3Uk3DEGj86sAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T22:32:32.790008Z"},"content_sha256":"9f59ea1cebcd9e6e5f13a75e9481f01c095f56343c8d36e31b0a19794ea101ff","schema_version":"1.0","event_id":"sha256:9f59ea1cebcd9e6e5f13a75e9481f01c095f56343c8d36e31b0a19794ea101ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OJ3UR442EZKJK4I67RT52MRPIX/bundle.json","state_url":"https://pith.science/pith/OJ3UR442EZKJK4I67RT52MRPIX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OJ3UR442EZKJK4I67RT52MRPIX/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":"2026-08-03T22:32:32Z","links":{"resolver":"https://pith.science/pith/OJ3UR442EZKJK4I67RT52MRPIX","bundle":"https://pith.science/pith/OJ3UR442EZKJK4I67RT52MRPIX/bundle.json","state":"https://pith.science/pith/OJ3UR442EZKJK4I67RT52MRPIX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OJ3UR442EZKJK4I67RT52MRPIX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OJ3UR442EZKJK4I67RT52MRPIX","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":"c558a6f5263a69d0cba86942c89dd6ce9e6373b28dc2a867c0bd62aeefcacc35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-21T19:36:18Z","title_canon_sha256":"6ab402e820122973e0bed1aea058097a28ed0a1c51d69d509b5a962e726f3d01"},"schema_version":"1.0","source":{"id":"1405.0290","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.0290","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"arxiv_version","alias_value":"1405.0290v5","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0290","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_12","alias_value":"OJ3UR442EZKJ","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_16","alias_value":"OJ3UR442EZKJK4I6","created_at":"2026-07-05T00:44:23Z"},{"alias_kind":"pith_short_8","alias_value":"OJ3UR442","created_at":"2026-07-05T00:44:23Z"}],"graph_snapshots":[{"event_id":"sha256:9f59ea1cebcd9e6e5f13a75e9481f01c095f56343c8d36e31b0a19794ea101ff","target":"graph","created_at":"2026-07-05T00:44:23Z","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/1405.0290/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\\sum_g(\\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\\ldots,p-1$, and $(\\frac{\\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-21T19:36:18Z","title":"New observations on primitive roots modulo primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0290","kind":"arxiv","version":5},"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:e130d9762d80533246149a403513c936befa9145015b742a5a135544ff74ae5e","target":"record","created_at":"2026-07-05T00:44:23Z","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":"c558a6f5263a69d0cba86942c89dd6ce9e6373b28dc2a867c0bd62aeefcacc35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-21T19:36:18Z","title_canon_sha256":"6ab402e820122973e0bed1aea058097a28ed0a1c51d69d509b5a962e726f3d01"},"schema_version":"1.0","source":{"id":"1405.0290","kind":"arxiv","version":5}},"canonical_sha256":"727748f39a265495711efc67dd322f45cf54c75f55568ea520409fa52a1f7800","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"727748f39a265495711efc67dd322f45cf54c75f55568ea520409fa52a1f7800","first_computed_at":"2026-07-05T00:44:23.342799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:44:23.342799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wwQfVu1qdElb3sEzwpbpyURXbWuBKGbnSmGs8tdDBOsA3LpdryxX474bDlnKAUVZsfdFUse/sViJTW1PFfaBBw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:44:23.343224Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.0290","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e130d9762d80533246149a403513c936befa9145015b742a5a135544ff74ae5e","sha256:9f59ea1cebcd9e6e5f13a75e9481f01c095f56343c8d36e31b0a19794ea101ff"],"state_sha256":"b29ec5dec15514672f3e8568f2ed43028649428158df2b2e09856af8dfaa0223"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i77PDwcE5CE2hv6gJreIjYYmVsqJcy1/rz94OVA445eofz/oOh5Exhm/n3UWUJZdenXzl5VqpJQ+fEAFOTcuDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T22:32:32.793420Z","bundle_sha256":"22bfde6f2fff9a5c5246726c733bfa236e4004a78ef7132d7f259719aac90b53"}}