{"paper":{"title":"Gaps between quadratic forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Siddharth Iyer","submitted_at":"2025-05-29T13:23:38Z","abstract_excerpt":"Let $\\triangle$ denote the integers represented by the quadratic form $x^2+xy+y^2$ and $\\square_{2}$ denote the numbers represented as a sum of two squares. For a non-zero integer $a$, let $S(\\triangle,\\square_{2},a)$ be the set of integers $n$ such that $n \\in \\triangle$, and $n + a \\in \\square_{2}$. We conduct a census of $S(\\triangle,\\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \\begin{align*} \\# S(\\triangle,\\square_{2},a)\\cap [x,x+H_{a}\\cdot x^{5/6}\\cdot \\log^{19}x] \\geq x^{5/6-\\varepsilon} \\end{align*} for large $x$. To derive this result and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23428","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.23428/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"}