On the minimal length of addition chains
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916682813931520 |
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| author | De Koninck, Jean-Marie Doyon, Nicolas Verreault, William |
| author_facet | De Koninck, Jean-Marie Doyon, Nicolas Verreault, William |
| contents | We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erdős which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the minimal length of addition chains De Koninck, Jean-Marie Doyon, Nicolas Verreault, William Number Theory Combinatorics 11B83 (Primary) 11Y55 (Secondary) We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erdős which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$ |
| title | On the minimal length of addition chains |
| topic | Number Theory Combinatorics 11B83 (Primary) 11Y55 (Secondary) |
| url | https://arxiv.org/abs/2504.07332 |