{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:BBNGFISXK7JKX6LCRFZN7HGZUM","short_pith_number":"pith:BBNGFISX","canonical_record":{"source":{"id":"2005.05214","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-05-11T16:03:21Z","cross_cats_sorted":[],"title_canon_sha256":"ba18e81de66daa108987ce399b4b51cc54cd329363aba540d910bc9d79543963","abstract_canon_sha256":"9addf5f43d7d1afc4a957c389ef8d3e8b7f89d628eb1d2eae2261545c9db10d1"},"schema_version":"1.0"},"canonical_sha256":"085a62a25757d2abf9628972df9cd9a32c47d748aaefdf37b5b4b1174a42e0c1","source":{"kind":"arxiv","id":"2005.05214","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.05214","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"arxiv_version","alias_value":"2005.05214v3","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.05214","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_12","alias_value":"BBNGFISXK7JK","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_16","alias_value":"BBNGFISXK7JKX6LC","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_8","alias_value":"BBNGFISX","created_at":"2026-07-05T06:15:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:BBNGFISXK7JKX6LCRFZN7HGZUM","target":"record","payload":{"canonical_record":{"source":{"id":"2005.05214","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-05-11T16:03:21Z","cross_cats_sorted":[],"title_canon_sha256":"ba18e81de66daa108987ce399b4b51cc54cd329363aba540d910bc9d79543963","abstract_canon_sha256":"9addf5f43d7d1afc4a957c389ef8d3e8b7f89d628eb1d2eae2261545c9db10d1"},"schema_version":"1.0"},"canonical_sha256":"085a62a25757d2abf9628972df9cd9a32c47d748aaefdf37b5b4b1174a42e0c1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:15:40.641529Z","signature_b64":"sUZ7vMfD9rghhj/OS4mWl3XYJB0gCW/mmKqAGKY/RKqfVpnVzQwfw8pSjUHPJ71Wbir4ah9HYz3ILnBROxHSCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"085a62a25757d2abf9628972df9cd9a32c47d748aaefdf37b5b4b1174a42e0c1","last_reissued_at":"2026-07-05T06:15:40.641026Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:15:40.641026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.05214","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:15:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ehCb4PHIILHmORaN8QzmrmRb+NqS3485PrXfOlbzW3vQnVoKN5YB3226qtNkzBweJgYmdm/AwRJcaMQ1th9pCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:27:26.172980Z"},"content_sha256":"1d98ad5c74ed65b01b01a96ec1a549052d87c903c684256840589b2e5e196edb","schema_version":"1.0","event_id":"sha256:1d98ad5c74ed65b01b01a96ec1a549052d87c903c684256840589b2e5e196edb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:BBNGFISXK7JKX6LCRFZN7HGZUM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a class of Lebesgue-Ramanujan-Nagell equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Azizul Hoque","submitted_at":"2020-05-11T16:03:21Z","abstract_excerpt":"We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\\geq 1, m\\geq 0$ and $n\\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\\not\\equiv 3 \\pmod 4$. We first solve this equation for prime $n$, under the condition $n\\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field $\\mathbb{Q}(\\sqrt{-cd})$. We then completely solve this equation for both $c$ and $d$ primes under the assumption that $\\gcd(n, h(-cd))=1$. We also completely solve this equation for $c=1$ and $d\\equiv1 \\pmod 4$, under the condition $\\gcd(n, h(-d))="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.05214","kind":"arxiv","version":3},"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/2005.05214/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":"2026-07-05T06:15:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kV7WW/mrVZAwjEw/kuv5Sbwcj5k5iicFwvBXCVsy3PlixA/9le4hVh32ZG/cRsbGkE5Ofj+Baqg2FHPUZxwhCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:27:26.173376Z"},"content_sha256":"6478904bfb642d1528249d2cb61d779a478b3b6a96b56408c2a640b78fe270fe","schema_version":"1.0","event_id":"sha256:6478904bfb642d1528249d2cb61d779a478b3b6a96b56408c2a640b78fe270fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/bundle.json","state_url":"https://pith.science/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/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":"2026-08-03T16:27:26Z","links":{"resolver":"https://pith.science/pith/BBNGFISXK7JKX6LCRFZN7HGZUM","bundle":"https://pith.science/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/bundle.json","state":"https://pith.science/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BBNGFISXK7JKX6LCRFZN7HGZUM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:BBNGFISXK7JKX6LCRFZN7HGZUM","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":"9addf5f43d7d1afc4a957c389ef8d3e8b7f89d628eb1d2eae2261545c9db10d1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-05-11T16:03:21Z","title_canon_sha256":"ba18e81de66daa108987ce399b4b51cc54cd329363aba540d910bc9d79543963"},"schema_version":"1.0","source":{"id":"2005.05214","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.05214","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"arxiv_version","alias_value":"2005.05214v3","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.05214","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_12","alias_value":"BBNGFISXK7JK","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_16","alias_value":"BBNGFISXK7JKX6LC","created_at":"2026-07-05T06:15:40Z"},{"alias_kind":"pith_short_8","alias_value":"BBNGFISX","created_at":"2026-07-05T06:15:40Z"}],"graph_snapshots":[{"event_id":"sha256:6478904bfb642d1528249d2cb61d779a478b3b6a96b56408c2a640b78fe270fe","target":"graph","created_at":"2026-07-05T06:15:40Z","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/2005.05214/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\\geq 1, m\\geq 0$ and $n\\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\\not\\equiv 3 \\pmod 4$. We first solve this equation for prime $n$, under the condition $n\\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field $\\mathbb{Q}(\\sqrt{-cd})$. We then completely solve this equation for both $c$ and $d$ primes under the assumption that $\\gcd(n, h(-cd))=1$. We also completely solve this equation for $c=1$ and $d\\equiv1 \\pmod 4$, under the condition $\\gcd(n, h(-d))=","authors_text":"Azizul Hoque","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-05-11T16:03:21Z","title":"On a class of Lebesgue-Ramanujan-Nagell equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.05214","kind":"arxiv","version":3},"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":"sha256:1d98ad5c74ed65b01b01a96ec1a549052d87c903c684256840589b2e5e196edb","target":"record","created_at":"2026-07-05T06:15:40Z","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":"9addf5f43d7d1afc4a957c389ef8d3e8b7f89d628eb1d2eae2261545c9db10d1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-05-11T16:03:21Z","title_canon_sha256":"ba18e81de66daa108987ce399b4b51cc54cd329363aba540d910bc9d79543963"},"schema_version":"1.0","source":{"id":"2005.05214","kind":"arxiv","version":3}},"canonical_sha256":"085a62a25757d2abf9628972df9cd9a32c47d748aaefdf37b5b4b1174a42e0c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"085a62a25757d2abf9628972df9cd9a32c47d748aaefdf37b5b4b1174a42e0c1","first_computed_at":"2026-07-05T06:15:40.641026Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:15:40.641026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sUZ7vMfD9rghhj/OS4mWl3XYJB0gCW/mmKqAGKY/RKqfVpnVzQwfw8pSjUHPJ71Wbir4ah9HYz3ILnBROxHSCw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:15:40.641529Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.05214","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d98ad5c74ed65b01b01a96ec1a549052d87c903c684256840589b2e5e196edb","sha256:6478904bfb642d1528249d2cb61d779a478b3b6a96b56408c2a640b78fe270fe"],"state_sha256":"4d754b5a0cd4e9dbef42f2abba43c98c85dff6a4dc462a0726e4824820ab275a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ivouIDafP77cLhy2gNQVvRwufPUzoD/zHqjSRYC2P3FJm4UYuN6tG4PitL6W+idRW/U0mRR4HPvxmHC10972Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:27:26.176030Z","bundle_sha256":"9a7026626272da03c8b5764b12b1391716372328a31614f2564f8325efbd1024"}}