{"paper":{"title":"Coefficient growth in square chains","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shawn Walker","submitted_at":"2019-02-27T18:55:33Z","abstract_excerpt":"Suppose $((\\cdots((x^{2}-c_{1})^{2}-c_{2})^{2}\\cdots)^{2}-c_{k-1})^{2}-c_{k}$ splits into linear factors over $\\mathbb{Z}$ and $c_{k}\\neq0$. We show that for each $j$ and each prime $p$, if $p\\leq2^{j-1}$ then $p$ divides $c_{j}$. Consequently, $$\\ln c_{j}>\\frac{1}{4}\\cdot2^{j}\\,\\,\\mathrm{for}\\,j\\geq5$$ If we also have $p\\equiv3\\,(\\mathrm{mod\\,4)}$ then $p^{2^{j-\\left\\lceil \\lg p\\right\\rceil }}$ divides $c_{j}$. Consequently, if $k\\geq3$, there exists some absolute constant $\\lambda>0$ so that, $$\\ln c_{j}>\\lambda k2^{j}\\mathrm{\\,\\,for\\,all\\,}j$$ These estimates argue against the possibility o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.11164","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"}