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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2603.03110 |
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| _version_ | 1866917330923028480 |
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| author | Lawton, Wayne M |
| author_facet | Lawton, Wayne M |
| contents | Assume that $n \geq 2$ and $B = (b_1,...,b_n)$ has distince integer entries $\geq 3.$ For $x > 0$ let $d_B(x) := (d_{b_1}(x),...,d_{b_n}(x))$ where $d_{b_i}(x) \in \{1,...,b_i-1\}$ is the leftmost digit in the base-$b_i$ positional notation representation of $x.$ We prove that if $d_B$ is surjective, then $\ln b_i$ and $\ln b_j$ are rationally independent whenever $i \neq j.$ We prove the converse for $n = 2,$ and for $n \geq 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Joint distribution of leftmost digits in positional notation and Schanuels's conjecture Lawton, Wayne M Number Theory 11A63, 11J81, 37A17 Assume that $n \geq 2$ and $B = (b_1,...,b_n)$ has distince integer entries $\geq 3.$ For $x > 0$ let $d_B(x) := (d_{b_1}(x),...,d_{b_n}(x))$ where $d_{b_i}(x) \in \{1,...,b_i-1\}$ is the leftmost digit in the base-$b_i$ positional notation representation of $x.$ We prove that if $d_B$ is surjective, then $\ln b_i$ and $\ln b_j$ are rationally independent whenever $i \neq j.$ We prove the converse for $n = 2,$ and for $n \geq 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers. |
| title | Joint distribution of leftmost digits in positional notation and Schanuels's conjecture |
| topic | Number Theory 11A63, 11J81, 37A17 |
| url | https://arxiv.org/abs/2603.03110 |