{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:6KELMZBUIG5RYLSO7NG5DOJQUC","short_pith_number":"pith:6KELMZBU","canonical_record":{"source":{"id":"0906.3533","kind":"arxiv","version":22},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-19T15:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"31d3045fdfa25f0e1e7c0aa263f86c95793fec6057f3a3616ffbb623bf24da8b","abstract_canon_sha256":"bc316183a1bfc9c9f0a70ad1b12a7cc5840c7a6b56dbec899903844d10ab3c60"},"schema_version":"1.0"},"canonical_sha256":"f288b6643441bb1c2e4efb4dd1b930a0be740da03a2d8672c1c9d7b69c36f3f5","source":{"kind":"arxiv","id":"0906.3533","version":22},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.3533","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"arxiv_version","alias_value":"0906.3533v22","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.3533","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"pith_short_12","alias_value":"6KELMZBUIG5R","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6KELMZBUIG5RYLSO","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6KELMZBU","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:6KELMZBUIG5RYLSO7NG5DOJQUC","target":"record","payload":{"canonical_record":{"source":{"id":"0906.3533","kind":"arxiv","version":22},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-19T15:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"31d3045fdfa25f0e1e7c0aa263f86c95793fec6057f3a3616ffbb623bf24da8b","abstract_canon_sha256":"bc316183a1bfc9c9f0a70ad1b12a7cc5840c7a6b56dbec899903844d10ab3c60"},"schema_version":"1.0"},"canonical_sha256":"f288b6643441bb1c2e4efb4dd1b930a0be740da03a2d8672c1c9d7b69c36f3f5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:32.304657Z","signature_b64":"AX0ZjbGv/JrWqvenH3SnX9Jc+914mbLtQ7Tpl6VfiNd8Nxps8zbyHzBCmKi/JO/0uK7+enRFXPwbt27CXBnuAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f288b6643441bb1c2e4efb4dd1b930a0be740da03a2d8672c1c9d7b69c36f3f5","last_reissued_at":"2026-05-18T03:07:32.304019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:32.304019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.3533","source_version":22,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BlVhdJ1Z7KR/V0+jXFRNoxKmfmwFppvQgN6iKyTgS8QX+FnRj44BfF3zRlOayj2nyCLKj0ql5wLMGZHcCEmOBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:27:57.131900Z"},"content_sha256":"259905ecf77a45032ba82a089127c369153ae3dcb469f72fc89cdb5efaebf451","schema_version":"1.0","event_id":"sha256:259905ecf77a45032ba82a089127c369153ae3dcb469f72fc89cdb5efaebf451"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:6KELMZBUIG5RYLSO7NG5DOJQUC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the distribution of Carmichael numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aran Nayebi","submitted_at":"2009-06-19T15:45:34Z","abstract_excerpt":"Erd\\H{o}s conjectured in 1956 that there are $x^{1-o(1)}$ Carmichael numbers up to $x$. Pomerance made this conjecture more precise and proposed that there are $x^{1-{\\frac{\\{1+o(1)\\}\\log\\log\\log x}{\\log\\log x}}}$ Carmichael numbers up to $x$. At the time, his data tables up to $25 \\cdot 10^{9}$ appeared to support his conjecture. However, Pinch extended this data and showed that up to $10^{21}$, Pomerance's conjecture did not appear well-supported. Thus, the purpose of this paper is two-fold. First, we build upon the work of Pomerance and others to present an alternate conjecture regarding th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3533","kind":"arxiv","version":22},"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":"2026-05-18T03:07:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nk+3qtZWewDY1+X6BCD+1Rh74SXgEdrW0EbyVwbB45VdgAXQIAxL/guEr8IY7D2Zxl58JJ3Bqx+8jazkphuwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:27:57.132655Z"},"content_sha256":"28f2d7637630861132863d5436fb15824c26ec0cb0679ca33a1a557c76f130b4","schema_version":"1.0","event_id":"sha256:28f2d7637630861132863d5436fb15824c26ec0cb0679ca33a1a557c76f130b4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/bundle.json","state_url":"https://pith.science/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/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":"2026-06-08T20:27:57Z","links":{"resolver":"https://pith.science/pith/6KELMZBUIG5RYLSO7NG5DOJQUC","bundle":"https://pith.science/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/bundle.json","state":"https://pith.science/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6KELMZBUIG5RYLSO7NG5DOJQUC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:6KELMZBUIG5RYLSO7NG5DOJQUC","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":"bc316183a1bfc9c9f0a70ad1b12a7cc5840c7a6b56dbec899903844d10ab3c60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-19T15:45:34Z","title_canon_sha256":"31d3045fdfa25f0e1e7c0aa263f86c95793fec6057f3a3616ffbb623bf24da8b"},"schema_version":"1.0","source":{"id":"0906.3533","kind":"arxiv","version":22}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.3533","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"arxiv_version","alias_value":"0906.3533v22","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.3533","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"pith_short_12","alias_value":"6KELMZBUIG5R","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6KELMZBUIG5RYLSO","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6KELMZBU","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:28f2d7637630861132863d5436fb15824c26ec0cb0679ca33a1a557c76f130b4","target":"graph","created_at":"2026-05-18T03:07:32Z","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":"Erd\\H{o}s conjectured in 1956 that there are $x^{1-o(1)}$ Carmichael numbers up to $x$. Pomerance made this conjecture more precise and proposed that there are $x^{1-{\\frac{\\{1+o(1)\\}\\log\\log\\log x}{\\log\\log x}}}$ Carmichael numbers up to $x$. At the time, his data tables up to $25 \\cdot 10^{9}$ appeared to support his conjecture. However, Pinch extended this data and showed that up to $10^{21}$, Pomerance's conjecture did not appear well-supported. Thus, the purpose of this paper is two-fold. First, we build upon the work of Pomerance and others to present an alternate conjecture regarding th","authors_text":"Aran Nayebi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-19T15:45:34Z","title":"On the distribution of Carmichael numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3533","kind":"arxiv","version":22},"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":"sha256:259905ecf77a45032ba82a089127c369153ae3dcb469f72fc89cdb5efaebf451","target":"record","created_at":"2026-05-18T03:07:32Z","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":"bc316183a1bfc9c9f0a70ad1b12a7cc5840c7a6b56dbec899903844d10ab3c60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-19T15:45:34Z","title_canon_sha256":"31d3045fdfa25f0e1e7c0aa263f86c95793fec6057f3a3616ffbb623bf24da8b"},"schema_version":"1.0","source":{"id":"0906.3533","kind":"arxiv","version":22}},"canonical_sha256":"f288b6643441bb1c2e4efb4dd1b930a0be740da03a2d8672c1c9d7b69c36f3f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f288b6643441bb1c2e4efb4dd1b930a0be740da03a2d8672c1c9d7b69c36f3f5","first_computed_at":"2026-05-18T03:07:32.304019Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:32.304019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AX0ZjbGv/JrWqvenH3SnX9Jc+914mbLtQ7Tpl6VfiNd8Nxps8zbyHzBCmKi/JO/0uK7+enRFXPwbt27CXBnuAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:32.304657Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.3533","source_kind":"arxiv","source_version":22}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:259905ecf77a45032ba82a089127c369153ae3dcb469f72fc89cdb5efaebf451","sha256:28f2d7637630861132863d5436fb15824c26ec0cb0679ca33a1a557c76f130b4"],"state_sha256":"a66cfdd083ef6ac89ca2e1aaf5c5efbc233427074036cb34c105803417ac97a1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GR9iEG/Y/1+j6JSM54FxPSswlgTIUmGhko816vEiEPZ8SoZ8BPLPR7md/xyjTfJeXabeMA7WYxxMBVfygG0PDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T20:27:57.136649Z","bundle_sha256":"1174a3872b6559790d5bb63c4a316ec24920b76b02d435d1e129395bf785bdd6"}}