{"paper":{"title":"Integer Complexity and Well-Ordering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Harry Altman","submitted_at":"2013-10-10T17:33:14Z","abstract_excerpt":"Define $\\|n\\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\\|n\\| \\ge 3\\log_3 n$ for all $n$. Define the defect of $n$, denoted $\\delta(n)$, to be $\\|n\\| - 3\\log_3 n$. In this paper, we consider the set $\\mathscr{D} := \\{\\delta(n): n \\ge 1 \\}$ of all defects. We show that as a subset of the real numbers, the set $\\mathscr{D}$ is well-ordered, of order type $\\omega^\\omega$. More specifically, for $k\\ge 1$ an integer, $\\mathscr{D}\\cap[0,k)$ has order type $\\omega^k$. We also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2894","kind":"arxiv","version":2},"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"}