{"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"}