The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910251597430784 |
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| author | van Doorn, Wouter Tang, Quanyu |
| author_facet | van Doorn, Wouter Tang, Quanyu |
| contents | Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots<n_k$ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of $v(k)$ and suggested the lower bound $v(k)\gg k!$. In this paper we give the first known improvement and show that there exists an absolute constant $c>0$ such that the inequality $$ v(k)\ge e^{c k^2} $$ holds for all positive integers $k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22083 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length van Doorn, Wouter Tang, Quanyu Number Theory Combinatorics Primary 11D68, Secondary 11B75 Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots<n_k$ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of $v(k)$ and suggested the lower bound $v(k)\gg k!$. In this paper we give the first known improvement and show that there exists an absolute constant $c>0$ such that the inequality $$ v(k)\ge e^{c k^2} $$ holds for all positive integers $k$. |
| title | The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length |
| topic | Number Theory Combinatorics Primary 11D68, Secondary 11B75 |
| url | https://arxiv.org/abs/2512.22083 |