{"paper":{"title":"A system of certain linear Diophantine equations on analogs of squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kota Saito, Yuya Kanado","submitted_at":"2022-05-24T17:34:54Z","abstract_excerpt":"This study investigates the existence of tuples $(k, \\ell, m)$ of integers such that all of $k$, $\\ell$, $m$, $k+\\ell$, $\\ell+m$, $m+k$, $k+\\ell+m$ belong to $S(\\alpha)$, where $S(\\alpha)$ is the set of all integers of the form $\\lfloor \\alpha n^2 \\rfloor$ for $n\\geq \\alpha^{-1/2}$ and $\\lfloor x\\rfloor$ denotes the integer part of $x$. We show that $T(\\alpha)$, the set of all such tuples, is infinite for all $\\alpha\\in (0,1)\\cap \\mathbb{Q}$ and for almost all $\\alpha\\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $\\alpha>0$ such that $T(\\alpha)$ is f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.12226","kind":"arxiv","version":4},"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/2205.12226/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"}