{"paper":{"title":"On congruences involving product of variables from short intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"M. Z. Garaev","submitted_at":"2017-01-25T00:51:16Z","abstract_excerpt":"We prove several results which imply the following consequences.\n  For any $\\varepsilon>0$ and any sufficiently large prime $p$, if $\\cI_1,\\ldots, \\cI_{13}$ are intervals of cardinalities $|\\cI_j|>p^{1/4+\\varepsilon}$ and $abc\\not\\equiv 0\\pmod p$, then the congruence $$ ax_1\\cdots x_6+bx_7\\cdots x_{13}\\equiv c\\pmod p $$ has a solution with $x_j\\in\\cI_j$.\n  There exists an absolute constant $n_0\\in\\N$ such that for any $0<\\varepsilon<1$ and any sufficiently large prime $p$, any quadratic residue $\\lambda$ modulo $p$ can be represented in the form $$ x_1\\cdots x_{n_0}\\equiv \\lambda\\pmod p,\\quad "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07119","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}