Powers in prime bases and a problem on central binomial coefficients
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arXiv
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| Natura: | Preprint |
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2026
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| author | Holdum, Sebastian Tim Klausen, Frederik Ravn Rasmussen, Peter Michael Reichstein |
| author_facet | Holdum, Sebastian Tim Klausen, Frederik Ravn Rasmussen, Peter Michael Reichstein |
| contents | It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do so we examine numbers of the form $α^{k}$ in base $p$, where $p$ is a prime and $(α, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$'s less than $a$ such that $(α^{k})_p$ contains less than $n$ digits greater than $\frac{p}{2}$. This is done by showing that every sequence of the form $\langle σ_t, \dots, σ_1,σ_0 \rangle$, where $0\leq σ_i<p$ for $i\geq 1$ and $σ_0$ is in the residue class generated by $α$ modulo $p$, occurs at specific places in the representation $(α^k)_p$ as $k$ varies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09510 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Powers in prime bases and a problem on central binomial coefficients Holdum, Sebastian Tim Klausen, Frederik Ravn Rasmussen, Peter Michael Reichstein Number Theory 11B65, 11A63, 11A07 It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do so we examine numbers of the form $α^{k}$ in base $p$, where $p$ is a prime and $(α, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$'s less than $a$ such that $(α^{k})_p$ contains less than $n$ digits greater than $\frac{p}{2}$. This is done by showing that every sequence of the form $\langle σ_t, \dots, σ_1,σ_0 \rangle$, where $0\leq σ_i<p$ for $i\geq 1$ and $σ_0$ is in the residue class generated by $α$ modulo $p$, occurs at specific places in the representation $(α^k)_p$ as $k$ varies. |
| title | Powers in prime bases and a problem on central binomial coefficients |
| topic | Number Theory 11B65, 11A63, 11A07 |
| url | https://arxiv.org/abs/2601.09510 |