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We show that for all $\\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\\alpha_2x_2^2+\\alpha_3x_3^2\\equiv 0 \\bmod{q}$ with $\\max\\{|x_1|,|x_2|,|x_3|\\}\\le N$ holds if $N\\ge q^{11/24+\\varepsilon}$ as $q$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2406.09778","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-06-14T07:27:19Z","cross_cats_sorted":[],"title_canon_sha256":"797c8282076e8fb94ba8fa7601b654f567810a254715e6105eeeadc2c2fc2c75","abstract_canon_sha256":"ec3ebb45304f5b0255851ab2f04993986fe5283260d749ee485e995ac1600ea8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:52:54.652555Z","signature_b64":"lsQUHHMyFUQrUSSXlPIdGYTmOwcx/ErG8oNm3EfmAC+cGRXctZS2mY3y4Rn9tNEzj2cOJRSy9236Mc54sMxmCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08723a0bab374810e8a4127b078f370265660b07892863dac2908a34c8358592","last_reissued_at":"2026-07-05T10:52:54.652104Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:52:54.652104Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Small Solutions of generic ternary quadratic congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aishik Chattopadhyay, Stephan Baier","submitted_at":"2024-06-14T07:27:19Z","abstract_excerpt":"We consider small solutions of quadratic congruences of the form $x_1^2+\\alpha_2x_2^2+\\alpha_3x_3^2\\equiv 0 \\bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\\alpha_2$ is arbitrary but fixed and $\\alpha_3$ is variable, and we assume that $(\\alpha_2\\alpha_3,q)=1$. We show that for all $\\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\\alpha_2x_2^2+\\alpha_3x_3^2\\equiv 0 \\bmod{q}$ with $\\max\\{|x_1|,|x_2|,|x_3|\\}\\le N$ holds if $N\\ge q^{11/24+\\varepsilon}$ as $q$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.09778","kind":"arxiv","version":5},"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/2406.09778/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2406.09778","created_at":"2026-07-05T10:52:54.652160+00:00"},{"alias_kind":"arxiv_version","alias_value":"2406.09778v5","created_at":"2026-07-05T10:52:54.652160+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.09778","created_at":"2026-07-05T10:52:54.652160+00:00"},{"alias_kind":"pith_short_12","alias_value":"BBZDUC5LG5EB","created_at":"2026-07-05T10:52:54.652160+00:00"},{"alias_kind":"pith_short_16","alias_value":"BBZDUC5LG5EBB2FE","created_at":"2026-07-05T10:52:54.652160+00:00"},{"alias_kind":"pith_short_8","alias_value":"BBZDUC5L","created_at":"2026-07-05T10:52:54.652160+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.02173","citing_title":"Data Diversification Methods In Alignment Enhance Math Performance In LLMs","ref_index":37,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ","json":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ.json","graph_json":"https://pith.science/api/pith-number/BBZDUC5LG5EBB2FECJ5QPDZXAJ/graph.json","events_json":"https://pith.science/api/pith-number/BBZDUC5LG5EBB2FECJ5QPDZXAJ/events.json","paper":"https://pith.science/paper/BBZDUC5L"},"agent_actions":{"view_html":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ","download_json":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ.json","view_paper":"https://pith.science/paper/BBZDUC5L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2406.09778&json=true","fetch_graph":"https://pith.science/api/pith-number/BBZDUC5LG5EBB2FECJ5QPDZXAJ/graph.json","fetch_events":"https://pith.science/api/pith-number/BBZDUC5LG5EBB2FECJ5QPDZXAJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ/action/storage_attestation","attest_author":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ/action/author_attestation","sign_citation":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ/action/citation_signature","submit_replication":"https://pith.science/pith/BBZDUC5LG5EBB2FECJ5QPDZXAJ/action/replication_record"}},"created_at":"2026-07-05T10:52:54.652160+00:00","updated_at":"2026-07-05T10:52:54.652160+00:00"}