{"paper":{"title":"Egyptian Fractions with Restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Elsholtz, Li-Li Jiang, Yong-Gao Chen","submitted_at":"2011-08-31T03:08:56Z","abstract_excerpt":"Let $T_o(k)$ denote the number of solutions of $\\sum_{i=1}^k\\frac 1{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1, p_2,..., p_t$, let $S(p_1, p_2,..., p_t)=\\{p_1^{\\alpha_1}...p_t^{\\alpha_t}\\mid \\alpha_i\\in \\mathbb{N}_0, i=1,2,..., t}$. Let $T_k(p_1,..., p_t)$ be the number of solutions $\\sum_{i=1}^{k}\\frac 1{x_i}=1$ with $1<x_1<x_2<...<x_{k}$ and $x_i\\in S(p_1, p_2,..., p_t)$. It is clear that if $T_k(p_1,..., p_t)\\not= 0$ for some $k$, then the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1.\n  In this paper we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.6118","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"}