Powers in prime bases and a problem on central binomial coefficients
Sebastian Tim Holdum, Frederik Ravn Klausen, Peter Michael Reichstein Rasmussen
TL;DR
The paper tackles the problem of whether the central binomial coefficients ${\binom{2n}{n}}$ are divisible by $4$ or $9$ for all large $n$, by focusing on the subfamily ${\binom{2^{k+1}}{2^k}}$ and analyzing carries via Kummer’s theorem. It develops a digit-based framework in prime bases: defining the in-base digits of powers $\alpha^k$, introducing the sets $\delta$, $\Lambda_k$, and the function ${\mathcal S}_p^n$, and proving that the sequences $\Lambda_k$ occur at prescribed positions in $(\alpha^s)_p$. This yields explicit bounds ${\mathcal S}_p^n(a) \le 8\,\log_p(a)^{n-1} a^{\log_p((p+1)/2)}$, which in turn imply that for any odd $m$, the set of $s$ with $m \nmid {\binom{2^{s+1}}{2^s}}$ has asymptotic density $0$, addressing conjectures for $m=9$ and suggesting a broader eventual-divisibility phenomenon. The work combines deep digit-pattern analysis with computational checks to strengthen and extend previous results on central binomial coefficient divisibility, providing a framework for further exploration of Erdős-type questions in this area.
Abstract
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.