{"paper":{"title":"On commuting integer matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Akshat Mudgal, Jonathan Chapman","submitted_at":"2025-04-22T12:34:14Z","abstract_excerpt":"Given $d, N \\in \\mathbb{N}$, we define $\\mathfrak{C}_d(N)$ to be the number of pairs of $d\\times d$ matrices $A,B$ with entries in $[-N,N] \\cap \\mathbb{Z}$ such that $AB = BA$. We prove that $$ N^{10} \\ll \\mathfrak{C}_3(N) \\ll N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further establish that $$ \\mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.15839","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/2504.15839/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"}