{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:NTU7KBGI5GCQAVGD247ABZEPHA","short_pith_number":"pith:NTU7KBGI","canonical_record":{"source":{"id":"0904.0473","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-02T21:46:53Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"4e19c563f49c95b1bfd5f88963e8d924d2e2638c183259aa4e0e5e8f532a680f","abstract_canon_sha256":"13e512503900397c303fb369c4709587577cf35add7b3cae8b87ecb9809cba0b"},"schema_version":"1.0"},"canonical_sha256":"6ce9f504c8e9850054c3d73e00e48f383adf967a24b30b98088ccdeee0ad5225","source":{"kind":"arxiv","id":"0904.0473","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.0473","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"arxiv_version","alias_value":"0904.0473v4","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.0473","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"pith_short_12","alias_value":"NTU7KBGI5GCQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NTU7KBGI5GCQAVGD","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NTU7KBGI","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:NTU7KBGI5GCQAVGD247ABZEPHA","target":"record","payload":{"canonical_record":{"source":{"id":"0904.0473","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-02T21:46:53Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"4e19c563f49c95b1bfd5f88963e8d924d2e2638c183259aa4e0e5e8f532a680f","abstract_canon_sha256":"13e512503900397c303fb369c4709587577cf35add7b3cae8b87ecb9809cba0b"},"schema_version":"1.0"},"canonical_sha256":"6ce9f504c8e9850054c3d73e00e48f383adf967a24b30b98088ccdeee0ad5225","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:37.369635Z","signature_b64":"0iuSVfuUWFQc3Lt9g2dS0SsA7+78e/KNy4AAAVApijThEkMQ4mc1t71nKZtZlTiYfIiWI+koCshgiHemZISpCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ce9f504c8e9850054c3d73e00e48f383adf967a24b30b98088ccdeee0ad5225","last_reissued_at":"2026-05-18T04:34:37.369147Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:37.369147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0904.0473","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:34:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JnYinCIPKYqURT91mUKIcYv+ng1hmwzDh78vTBoX1CaUmCpei7A1guIIlEvdjIpK44WY/wLrZo3roo2COEYJCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T09:41:27.020720Z"},"content_sha256":"0fa1ce3828193a52ae23528ab5fe07dd2a587b6a713bcb3a308ff39ea864f2aa","schema_version":"1.0","event_id":"sha256:0fa1ce3828193a52ae23528ab5fe07dd2a587b6a713bcb3a308ff39ea864f2aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:NTU7KBGI5GCQAVGD247ABZEPHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Prime chains and Pratt trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Florian Luca, Kevin Ford, Sergei V. Konyagin","submitted_at":"2009-04-02T21:46:53Z","abstract_excerpt":"We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\\equiv 1\\pmod{p_j} for each j. We give estimates for the number of chains with p_k\\le x (k variable), and the number of chains with p_1=p and p_k \\le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\\ge c\\log\\log p and H(p)\\le (\\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any \\epsilon>0, c=e-\\epsilon assuming the Elliott-Halbersta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0473","kind":"arxiv","version":4},"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":"2026-05-18T04:34:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dRUD4udLRJKuzjVbDIT6HTw/wG1ALBxyJGP5F68mipAkBWqeQcpnv3nmnin4VgqEXrRk3DwhYaXk8kWFcWGZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T09:41:27.021170Z"},"content_sha256":"3f7ee1571348c7bf20350147376eea92300b9a0882b11d7ed4c1e1768d2dc188","schema_version":"1.0","event_id":"sha256:3f7ee1571348c7bf20350147376eea92300b9a0882b11d7ed4c1e1768d2dc188"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NTU7KBGI5GCQAVGD247ABZEPHA/bundle.json","state_url":"https://pith.science/pith/NTU7KBGI5GCQAVGD247ABZEPHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NTU7KBGI5GCQAVGD247ABZEPHA/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":"2026-05-25T09:41:27Z","links":{"resolver":"https://pith.science/pith/NTU7KBGI5GCQAVGD247ABZEPHA","bundle":"https://pith.science/pith/NTU7KBGI5GCQAVGD247ABZEPHA/bundle.json","state":"https://pith.science/pith/NTU7KBGI5GCQAVGD247ABZEPHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NTU7KBGI5GCQAVGD247ABZEPHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NTU7KBGI5GCQAVGD247ABZEPHA","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":"13e512503900397c303fb369c4709587577cf35add7b3cae8b87ecb9809cba0b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-02T21:46:53Z","title_canon_sha256":"4e19c563f49c95b1bfd5f88963e8d924d2e2638c183259aa4e0e5e8f532a680f"},"schema_version":"1.0","source":{"id":"0904.0473","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.0473","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"arxiv_version","alias_value":"0904.0473v4","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.0473","created_at":"2026-05-18T04:34:37Z"},{"alias_kind":"pith_short_12","alias_value":"NTU7KBGI5GCQ","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NTU7KBGI5GCQAVGD","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NTU7KBGI","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:3f7ee1571348c7bf20350147376eea92300b9a0882b11d7ed4c1e1768d2dc188","target":"graph","created_at":"2026-05-18T04:34:37Z","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 study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\\equiv 1\\pmod{p_j} for each j. We give estimates for the number of chains with p_k\\le x (k variable), and the number of chains with p_1=p and p_k \\le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\\ge c\\log\\log p and H(p)\\le (\\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any \\epsilon>0, c=e-\\epsilon assuming the Elliott-Halbersta","authors_text":"Florian Luca, Kevin Ford, Sergei V. Konyagin","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-02T21:46:53Z","title":"Prime chains and Pratt trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0473","kind":"arxiv","version":4},"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":"sha256:0fa1ce3828193a52ae23528ab5fe07dd2a587b6a713bcb3a308ff39ea864f2aa","target":"record","created_at":"2026-05-18T04:34:37Z","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":"13e512503900397c303fb369c4709587577cf35add7b3cae8b87ecb9809cba0b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-02T21:46:53Z","title_canon_sha256":"4e19c563f49c95b1bfd5f88963e8d924d2e2638c183259aa4e0e5e8f532a680f"},"schema_version":"1.0","source":{"id":"0904.0473","kind":"arxiv","version":4}},"canonical_sha256":"6ce9f504c8e9850054c3d73e00e48f383adf967a24b30b98088ccdeee0ad5225","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ce9f504c8e9850054c3d73e00e48f383adf967a24b30b98088ccdeee0ad5225","first_computed_at":"2026-05-18T04:34:37.369147Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:37.369147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0iuSVfuUWFQc3Lt9g2dS0SsA7+78e/KNy4AAAVApijThEkMQ4mc1t71nKZtZlTiYfIiWI+koCshgiHemZISpCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:37.369635Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.0473","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0fa1ce3828193a52ae23528ab5fe07dd2a587b6a713bcb3a308ff39ea864f2aa","sha256:3f7ee1571348c7bf20350147376eea92300b9a0882b11d7ed4c1e1768d2dc188"],"state_sha256":"9bae8bc2bdc3a8657c6f93c0f11f53c1e50d2bc4df5e32e531a07d1630ca1209"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vGY2TPUqjOM2hRaO58CcUC2dE3ymK+bD0e/takFonFQAmjvvtAzrlJSH9AOEGSBajX2pRw9ryGvvNpQKzj3BBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T09:41:27.024608Z","bundle_sha256":"c0c8ddda04620a4bcb1ff91f00456556f19fdb3d26327e10f67e9c13f035074f"}}