{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:Y2BDXUPA4KI4L5MNXPOAOIWEJB","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":"9e053175cadd35c022a541d180a115767967a6cc54b9cad1ef71b29b4dd5a8e9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-06-03T16:12:25Z","title_canon_sha256":"a584adba8c98514fa499a65d930add6da58a605a6f793184330d38bacc77a8d3"},"schema_version":"1.0","source":{"id":"2106.01964","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.01964","created_at":"2026-07-05T03:30:39Z"},{"alias_kind":"arxiv_version","alias_value":"2106.01964v2","created_at":"2026-07-05T03:30:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.01964","created_at":"2026-07-05T03:30:39Z"},{"alias_kind":"pith_short_12","alias_value":"Y2BDXUPA4KI4","created_at":"2026-07-05T03:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"Y2BDXUPA4KI4L5MN","created_at":"2026-07-05T03:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"Y2BDXUPA","created_at":"2026-07-05T03:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:e9e09bccf15e23c7bf3908823079d65b81fb3143994a3c04bdc78265374e6635","target":"graph","created_at":"2026-07-05T03:30:39Z","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/2106.01964/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\\gcd(x,y)=1$. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.","authors_text":"Azizul Hoque, Kalyan Chakraborty","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-06-03T16:12:25Z","title":"On the Diophantine equation $dx^2+p^{2a}q^{2b}=4y^p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.01964","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":"sha256:50c962661882137f8ebf1f121b87fe08588aa5ba0b6b18f57204264120fba41a","target":"record","created_at":"2026-07-05T03:30:39Z","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":"9e053175cadd35c022a541d180a115767967a6cc54b9cad1ef71b29b4dd5a8e9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-06-03T16:12:25Z","title_canon_sha256":"a584adba8c98514fa499a65d930add6da58a605a6f793184330d38bacc77a8d3"},"schema_version":"1.0","source":{"id":"2106.01964","kind":"arxiv","version":2}},"canonical_sha256":"c6823bd1e0e291c5f58dbbdc0722c44844b5e5bb0cbcbb02a9309490345ed93d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6823bd1e0e291c5f58dbbdc0722c44844b5e5bb0cbcbb02a9309490345ed93d","first_computed_at":"2026-07-05T03:30:39.840730Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:30:39.840730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y+/fO37kFx3/S2a7N+oWiKhiR/An4oPrbJkbb4FirNHv9+2c6Ccaon/Jw9ZDxbEGWZU1qxKlydIcMMRg/aYDAg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:30:39.841204Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.01964","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50c962661882137f8ebf1f121b87fe08588aa5ba0b6b18f57204264120fba41a","sha256:e9e09bccf15e23c7bf3908823079d65b81fb3143994a3c04bdc78265374e6635"],"state_sha256":"6aa7b13fd33b4b9d6c878a0481584c011b3f7377b51fb80f6561ac8776a8ccc4"}