{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:U4TFTRJQNFC45J36LK3W7ZRAI6","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":"fc10a73e9b8ad401dfcba482c5a262ab3ba288e05cab309f5442e4effafadaab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-05T09:04:39Z","title_canon_sha256":"58ecfa2e39e7aa18a7c287a8556a243b0231e82c970579f83f6a02fcdfb91e14"},"schema_version":"1.0","source":{"id":"1810.02573","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.02573","created_at":"2026-05-17T23:51:49Z"},{"alias_kind":"arxiv_version","alias_value":"1810.02573v2","created_at":"2026-05-17T23:51:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.02573","created_at":"2026-05-17T23:51:49Z"},{"alias_kind":"pith_short_12","alias_value":"U4TFTRJQNFC4","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"U4TFTRJQNFC45J36","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"U4TFTRJQ","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:13078c0529a7eee666dcd20eb262833c5a98ec63e6462e9d7a741300aadb2901","target":"graph","created_at":"2026-05-17T23:51:49Z","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 obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.","authors_text":"Igor E. Shparlinski, Kam Hung Yau, Marc Munsch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-05T09:04:39Z","title":"Smooth squarefree and square-full integers in arithmetic progressions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.02573","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":"sha256:53635894b437b76db7f5d35ce3df9f53898ee3f1884cd14b6dfbddca099eeef6","target":"record","created_at":"2026-05-17T23:51:49Z","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":"fc10a73e9b8ad401dfcba482c5a262ab3ba288e05cab309f5442e4effafadaab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-05T09:04:39Z","title_canon_sha256":"58ecfa2e39e7aa18a7c287a8556a243b0231e82c970579f83f6a02fcdfb91e14"},"schema_version":"1.0","source":{"id":"1810.02573","kind":"arxiv","version":2}},"canonical_sha256":"a72659c5306945cea77e5ab76fe62047b41baf731c4985c554e3b2bb1617a62b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a72659c5306945cea77e5ab76fe62047b41baf731c4985c554e3b2bb1617a62b","first_computed_at":"2026-05-17T23:51:49.072424Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:49.072424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5nQOcmRttd8ixBxvRfoPPi9nmGBCVIcuJzrafccyE9KeKREPkE+4YoL4VOKyyrLxxDHgcUmGkn3re68roSsEAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:49.072944Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.02573","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53635894b437b76db7f5d35ce3df9f53898ee3f1884cd14b6dfbddca099eeef6","sha256:13078c0529a7eee666dcd20eb262833c5a98ec63e6462e9d7a741300aadb2901"],"state_sha256":"9c2471e0e25fef650e2d85928fa5df2f0586fe25712105a0664a7e40305d01ca"}