{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:CFZQ35RU7G6U6O2NDQAH3EGDPO","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":"7aad8ef1754a15ca148b8a541a382b8044d58cbaec60ad10b6f2812167d5a017","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-23T13:21:57Z","title_canon_sha256":"1b2752f053455ab47237012ed1ad7e12aac08c109068e4ac04f7eafa94d58997"},"schema_version":"1.0","source":{"id":"1908.08816","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08816","created_at":"2026-07-05T01:48:06Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08816v3","created_at":"2026-07-05T01:48:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08816","created_at":"2026-07-05T01:48:06Z"},{"alias_kind":"pith_short_12","alias_value":"CFZQ35RU7G6U","created_at":"2026-07-05T01:48:06Z"},{"alias_kind":"pith_short_16","alias_value":"CFZQ35RU7G6U6O2N","created_at":"2026-07-05T01:48:06Z"},{"alias_kind":"pith_short_8","alias_value":"CFZQ35RU","created_at":"2026-07-05T01:48:06Z"}],"graph_snapshots":[{"event_id":"sha256:c62cf624918a5f63419345c1ccaf26b58599becd991711583c4103c21077fbfd","target":"graph","created_at":"2026-07-05T01:48:06Z","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/1908.08816/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$","authors_text":"Jori Merikoski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-23T13:21:57Z","title":"On the largest prime factor of $n^2+1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08816","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":"sha256:6d21f8ac32ef61315a791969b8a1319b6b87bed3d4f3463060c93a13b0418c80","target":"record","created_at":"2026-07-05T01:48:06Z","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":"7aad8ef1754a15ca148b8a541a382b8044d58cbaec60ad10b6f2812167d5a017","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-23T13:21:57Z","title_canon_sha256":"1b2752f053455ab47237012ed1ad7e12aac08c109068e4ac04f7eafa94d58997"},"schema_version":"1.0","source":{"id":"1908.08816","kind":"arxiv","version":3}},"canonical_sha256":"11730df634f9bd4f3b4d1c007d90c37b8f31e6753e3f134d293117d01aacb8a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11730df634f9bd4f3b4d1c007d90c37b8f31e6753e3f134d293117d01aacb8a8","first_computed_at":"2026-07-05T01:48:06.313943Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:48:06.313943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UJU855BWB+PqmCKHAgxf8jaDITvM+KR8JaWxrUTFIQqrb463GpL8R9y1lwRmTdSjk6BG3PdHVelPeNEQ6lY9Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:48:06.314289Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08816","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d21f8ac32ef61315a791969b8a1319b6b87bed3d4f3463060c93a13b0418c80","sha256:c62cf624918a5f63419345c1ccaf26b58599becd991711583c4103c21077fbfd"],"state_sha256":"7ccb6c175f8ee43deb8285e54787355c1b5774a8067dfbe34a770981e0b1aad7"}