{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:7AZSZ6UIHVH7NOVJKIDGEBORKZ","short_pith_number":"pith:7AZSZ6UI","canonical_record":{"source":{"id":"0908.0554","kind":"arxiv","version":25},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"befe3ec0e86afd28abf5c899792f93fc29099adc52dd623c9b634178d18a799d","abstract_canon_sha256":"085d7410afbb86bf9bb04e6c689d8d9ea3309040e683ecef400a9426bc420728"},"schema_version":"1.0"},"canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","source":{"kind":"arxiv","id":"0908.0554","version":25},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0554","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0554v25","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0554","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"pith_short_12","alias_value":"7AZSZ6UIHVH7","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"7AZSZ6UIHVH7NOVJ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"7AZSZ6UI","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:7AZSZ6UIHVH7NOVJKIDGEBORKZ","target":"record","payload":{"canonical_record":{"source":{"id":"0908.0554","kind":"arxiv","version":25},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"befe3ec0e86afd28abf5c899792f93fc29099adc52dd623c9b634178d18a799d","abstract_canon_sha256":"085d7410afbb86bf9bb04e6c689d8d9ea3309040e683ecef400a9426bc420728"},"schema_version":"1.0"},"canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:05.545226Z","signature_b64":"NwEG6IC79mGBrew0P1pRUEIj8CcXIa6GOztLYWrTz1GBPlxWRCn3AFdkCy7zc43s6nGzc0VN2xiuwPDlOKB1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","last_reissued_at":"2026-05-18T04:20:05.544807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:05.544807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.0554","source_version":25,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:20:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z0+eOTYeurDmSxmd5YNOuX3nD1HVcRYD/ToV5l0IGFQtLE4zLxybVn3JFbfyLzdzDYuankd280MrvpMxPncRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:02:51.381182Z"},"content_sha256":"12701e6287a31b572f5f660c377401efdd23a25eb39ecf32c80d70318bc93920","schema_version":"1.0","event_id":"sha256:12701e6287a31b572f5f660c377401efdd23a25eb39ecf32c80d70318bc93920"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:7AZSZ6UIHVH7NOVJKIDGEBORKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On integers as the sum of a prime and a $k$-th power","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.NT","authors_text":"Aran Nayebi","submitted_at":"2009-08-05T18:33:44Z","abstract_excerpt":"Let $\\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\\{n \\le X, n \\in I_k, n\\text{not a sum of a prime and a $k$-th power}\\}|.\n  Hardy and Littlewood conjectured that for $k = 2$ and $k=3$, E_k(X) \\ll_{k} 1. In this note we present an alternative approach grounded in the theory of Diophantine equations towards a proof of the conjecture for all $k \\ge 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0554","kind":"arxiv","version":25},"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":"2026-05-18T04:20:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RiVdhn9m+suuoMJdPOOV+Ji3lZiSlKVe0N27+RXJBSY47JgLHZ8zWm1vQGdvnLNhKn78kgv6bEHPP8x9XLKJBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:02:51.381564Z"},"content_sha256":"a33ec928b283ff4a22c0582ebe57a40a3894e369f144b646d47b96f6f9176e49","schema_version":"1.0","event_id":"sha256:a33ec928b283ff4a22c0582ebe57a40a3894e369f144b646d47b96f6f9176e49"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/bundle.json","state_url":"https://pith.science/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/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":"2026-06-08T17:02:51Z","links":{"resolver":"https://pith.science/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ","bundle":"https://pith.science/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/bundle.json","state":"https://pith.science/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7AZSZ6UIHVH7NOVJKIDGEBORKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:7AZSZ6UIHVH7NOVJKIDGEBORKZ","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":"085d7410afbb86bf9bb04e6c689d8d9ea3309040e683ecef400a9426bc420728","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","title_canon_sha256":"befe3ec0e86afd28abf5c899792f93fc29099adc52dd623c9b634178d18a799d"},"schema_version":"1.0","source":{"id":"0908.0554","kind":"arxiv","version":25}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0554","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0554v25","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0554","created_at":"2026-05-18T04:20:05Z"},{"alias_kind":"pith_short_12","alias_value":"7AZSZ6UIHVH7","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"7AZSZ6UIHVH7NOVJ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"7AZSZ6UI","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a33ec928b283ff4a22c0582ebe57a40a3894e369f144b646d47b96f6f9176e49","target":"graph","created_at":"2026-05-18T04:20:05Z","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":"Let $\\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\\{n \\le X, n \\in I_k, n\\text{not a sum of a prime and a $k$-th power}\\}|.\n  Hardy and Littlewood conjectured that for $k = 2$ and $k=3$, E_k(X) \\ll_{k} 1. In this note we present an alternative approach grounded in the theory of Diophantine equations towards a proof of the conjecture for all $k \\ge 2$.","authors_text":"Aran Nayebi","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","title":"On integers as the sum of a prime and a $k$-th power"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0554","kind":"arxiv","version":25},"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":"sha256:12701e6287a31b572f5f660c377401efdd23a25eb39ecf32c80d70318bc93920","target":"record","created_at":"2026-05-18T04:20:05Z","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":"085d7410afbb86bf9bb04e6c689d8d9ea3309040e683ecef400a9426bc420728","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","title_canon_sha256":"befe3ec0e86afd28abf5c899792f93fc29099adc52dd623c9b634178d18a799d"},"schema_version":"1.0","source":{"id":"0908.0554","kind":"arxiv","version":25}},"canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","first_computed_at":"2026-05-18T04:20:05.544807Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:20:05.544807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NwEG6IC79mGBrew0P1pRUEIj8CcXIa6GOztLYWrTz1GBPlxWRCn3AFdkCy7zc43s6nGzc0VN2xiuwPDlOKB1Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:20:05.545226Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.0554","source_kind":"arxiv","source_version":25}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12701e6287a31b572f5f660c377401efdd23a25eb39ecf32c80d70318bc93920","sha256:a33ec928b283ff4a22c0582ebe57a40a3894e369f144b646d47b96f6f9176e49"],"state_sha256":"37b0c07f0ddca7bf5cb5fec30597da9ec0b077ad92212918580afc9fb115a100"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Kbi+UuGJVGCbYKsbtHs1r3uEshpiryLRMqbrKisBk19bVX0VYiYZeAwGMr0QBXfBVI26b/+IcIBft95fdAHBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:02:51.383782Z","bundle_sha256":"6cd8ebeded4da8f3d0f0665119bb68d266dd2da83b3c8b21352f3418df83a8a1"}}