{"paper":{"title":"Integer Complexity: Representing Numbers of Bounded Defect","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Harry Altman","submitted_at":"2016-03-19T18:00:33Z","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 $n$. Based on this, this author and Zelinsky defined the \"defect\" of $n$, $\\delta(n):=\\|n\\|-3\\log_3 n$, and this author showed that the set of all defects is a well-ordered subset of the real numbers. This was accomplished by showing that for a fixed real number $r$, there is a finite set $S$ of polynomials called \"low-defect polynomials\" such that for any $n$ with $\\delta(n)<r$, $n$ ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06122","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"}