A lower bound for the number of Egyptian fractions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908535182327808 |
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| author | Bettin, Sandro Grenié, Loïc Molteni, Giuseppe Sanna, Carlo |
| author_facet | Bettin, Sandro Grenié, Loïc Molteni, Giuseppe Sanna, Carlo |
| contents | An Egyptian fraction is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds $$ \frac{\ln(|E_N|)}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , $$ where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition $\ln_k N\geq k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $|E_N|$ for large positive integers $N$, and we give a table of $|E_N|$ for $N$ up to $154$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lower bound for the number of Egyptian fractions Bettin, Sandro Grenié, Loïc Molteni, Giuseppe Sanna, Carlo Number Theory 11D68 (Primary) 11B99 (Secondary) An Egyptian fraction is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds $$ \frac{\ln(|E_N|)}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , $$ where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition $\ln_k N\geq k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $|E_N|$ for large positive integers $N$, and we give a table of $|E_N|$ for $N$ up to $154$. |
| title | A lower bound for the number of Egyptian fractions |
| topic | Number Theory 11D68 (Primary) 11B99 (Secondary) |
| url | https://arxiv.org/abs/2509.10030 |