{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:5QVBO2FDKAFEZ4JKO7BFBQUZ4N","short_pith_number":"pith:5QVBO2FD","canonical_record":{"source":{"id":"1809.08431","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T12:44:45Z","cross_cats_sorted":[],"title_canon_sha256":"787bb0bddf851170079a8cc8872f9a829cd35edb3ae9b69509befd986a931792","abstract_canon_sha256":"4a155aecb6a170b7101fd12fcdd401a116d1d3c737739a51204b87e0e7ec51c8"},"schema_version":"1.0"},"canonical_sha256":"ec2a1768a3500a4cf12a77c250c299e35f5664022d1265eea5bce3d0a8f1d8f9","source":{"kind":"arxiv","id":"1809.08431","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.08431","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"arxiv_version","alias_value":"1809.08431v2","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08431","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"pith_short_12","alias_value":"5QVBO2FDKAFE","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"5QVBO2FDKAFEZ4JK","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"5QVBO2FD","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:5QVBO2FDKAFEZ4JKO7BFBQUZ4N","target":"record","payload":{"canonical_record":{"source":{"id":"1809.08431","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T12:44:45Z","cross_cats_sorted":[],"title_canon_sha256":"787bb0bddf851170079a8cc8872f9a829cd35edb3ae9b69509befd986a931792","abstract_canon_sha256":"4a155aecb6a170b7101fd12fcdd401a116d1d3c737739a51204b87e0e7ec51c8"},"schema_version":"1.0"},"canonical_sha256":"ec2a1768a3500a4cf12a77c250c299e35f5664022d1265eea5bce3d0a8f1d8f9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:55.110385Z","signature_b64":"WYu3rTTKAeox8HXycOgg+O9hO2Hia7AeZ/TUJKC5PobLVx4W9U2mk52PutUMTwkz8PG3dABSijibAT7nAKzIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec2a1768a3500a4cf12a77c250c299e35f5664022d1265eea5bce3d0a8f1d8f9","last_reissued_at":"2026-05-17T23:46:55.109769Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:55.109769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.08431","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:46:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cYB6RXKyiFZVjpj9pIQghuv0zXqAVMwl+5RHhSUB0YrqO0lotHtp+SuOScU1vhP8gz3NuvEcVps8HwbjH7vKDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:57:29.678419Z"},"content_sha256":"d7f1410435bf5409c96a76ffd2944766ab113162f7c74a1bebf89ee6c86060fd","schema_version":"1.0","event_id":"sha256:d7f1410435bf5409c96a76ffd2944766ab113162f7c74a1bebf89ee6c86060fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:5QVBO2FDKAFEZ4JKO7BFBQUZ4N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Sha, Min-Soo Kim, Pieter Moree, Su Hu","submitted_at":"2018-09-22T12:44:45Z","abstract_excerpt":"In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08431","kind":"arxiv","version":2},"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":"2026-05-17T23:46:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8yIBlHrNW7xYQRrSs4JKRk0vqsjh+VlOiFg/kB7rbMjUN/HuBkkbnGaJMO3wzjUazs+bad93fNymlMA9mCjoBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:57:29.678828Z"},"content_sha256":"e4eebe97ebf8eb2383c0f5ffa9481218f7db674d40239cfc4cfc48d474c339eb","schema_version":"1.0","event_id":"sha256:e4eebe97ebf8eb2383c0f5ffa9481218f7db674d40239cfc4cfc48d474c339eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/bundle.json","state_url":"https://pith.science/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/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":"2026-05-25T19:57:29Z","links":{"resolver":"https://pith.science/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N","bundle":"https://pith.science/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/bundle.json","state":"https://pith.science/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5QVBO2FDKAFEZ4JKO7BFBQUZ4N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:5QVBO2FDKAFEZ4JKO7BFBQUZ4N","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":"4a155aecb6a170b7101fd12fcdd401a116d1d3c737739a51204b87e0e7ec51c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T12:44:45Z","title_canon_sha256":"787bb0bddf851170079a8cc8872f9a829cd35edb3ae9b69509befd986a931792"},"schema_version":"1.0","source":{"id":"1809.08431","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.08431","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"arxiv_version","alias_value":"1809.08431v2","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08431","created_at":"2026-05-17T23:46:55Z"},{"alias_kind":"pith_short_12","alias_value":"5QVBO2FDKAFE","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"5QVBO2FDKAFEZ4JK","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"5QVBO2FD","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:e4eebe97ebf8eb2383c0f5ffa9481218f7db674d40239cfc4cfc48d474c339eb","target":"graph","created_at":"2026-05-17T23:46:55Z","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":"In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely ma","authors_text":"Min Sha, Min-Soo Kim, Pieter Moree, Su Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T12:44:45Z","title":"Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08431","kind":"arxiv","version":2},"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:d7f1410435bf5409c96a76ffd2944766ab113162f7c74a1bebf89ee6c86060fd","target":"record","created_at":"2026-05-17T23:46:55Z","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":"4a155aecb6a170b7101fd12fcdd401a116d1d3c737739a51204b87e0e7ec51c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T12:44:45Z","title_canon_sha256":"787bb0bddf851170079a8cc8872f9a829cd35edb3ae9b69509befd986a931792"},"schema_version":"1.0","source":{"id":"1809.08431","kind":"arxiv","version":2}},"canonical_sha256":"ec2a1768a3500a4cf12a77c250c299e35f5664022d1265eea5bce3d0a8f1d8f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec2a1768a3500a4cf12a77c250c299e35f5664022d1265eea5bce3d0a8f1d8f9","first_computed_at":"2026-05-17T23:46:55.109769Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:55.109769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WYu3rTTKAeox8HXycOgg+O9hO2Hia7AeZ/TUJKC5PobLVx4W9U2mk52PutUMTwkz8PG3dABSijibAT7nAKzIDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:55.110385Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.08431","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d7f1410435bf5409c96a76ffd2944766ab113162f7c74a1bebf89ee6c86060fd","sha256:e4eebe97ebf8eb2383c0f5ffa9481218f7db674d40239cfc4cfc48d474c339eb"],"state_sha256":"ae9645ba044410f9e5f0db33dac7f2431b4da0f37277ccb1360992ee311c066c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yxvXdqXQDNbGgmSJbqRERnX/aGRH6mXB1ehFl1rNgh6PTgqN/rQj6+ykttrHgBfJGANDxkJ6tuFJqswfQIu6Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T19:57:29.682215Z","bundle_sha256":"b0761c570acc183a3bcb74e0b1623ed99380fba10a852bc0dc0b0ea26fd65eb1"}}