{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QXNC4PEGMRWHLY7OWMB6J3ITYI","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":"4789f8813cc3aa1d9393968ee70ab86fcb52366e32b4cf976f65818cca7e26a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-20T03:30:31Z","title_canon_sha256":"1e157eeab7a2da9e8e3b6f275e083e21822527b4cad83e414f7a4ae74a61a801"},"schema_version":"1.0","source":{"id":"1408.4515","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.4515","created_at":"2026-05-18T02:03:48Z"},{"alias_kind":"arxiv_version","alias_value":"1408.4515v2","created_at":"2026-05-18T02:03:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.4515","created_at":"2026-05-18T02:03:48Z"},{"alias_kind":"pith_short_12","alias_value":"QXNC4PEGMRWH","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QXNC4PEGMRWHLY7O","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QXNC4PEG","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:444608decef3034c5c89667979804aaa0398a59fab1ea8b97546b6da518a3b75","target":"graph","created_at":"2026-05-18T02:03:48Z","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 show that for any $\\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \\varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.","authors_text":"Glyn Harman, Igor E. Shparlinski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-20T03:30:31Z","title":"Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4515","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:c61da0f9077dda0cca80d71b7138d94eade421de8e11a11b56884ebce9c2eaf3","target":"record","created_at":"2026-05-18T02:03:48Z","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":"4789f8813cc3aa1d9393968ee70ab86fcb52366e32b4cf976f65818cca7e26a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-20T03:30:31Z","title_canon_sha256":"1e157eeab7a2da9e8e3b6f275e083e21822527b4cad83e414f7a4ae74a61a801"},"schema_version":"1.0","source":{"id":"1408.4515","kind":"arxiv","version":2}},"canonical_sha256":"85da2e3c86646c75e3eeb303e4ed13c21f28abd2000753e82b8b196f3909cdcd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85da2e3c86646c75e3eeb303e4ed13c21f28abd2000753e82b8b196f3909cdcd","first_computed_at":"2026-05-18T02:03:48.300549Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:48.300549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rCQNfcVzfpYVFuXa2j8uVfSAsaVnuUgdUHKcSxS3JU4z6MD0931xKxx9pwxbaBpTB/ubNDHs9ptMKW0JfOGMAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:48.301154Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.4515","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c61da0f9077dda0cca80d71b7138d94eade421de8e11a11b56884ebce9c2eaf3","sha256:444608decef3034c5c89667979804aaa0398a59fab1ea8b97546b6da518a3b75"],"state_sha256":"6c734f39a39e1155bcf472d4d10b04a88f28c1b992c8d7b0cf34bc2dc76da3ae"}