Physical Sciences and Mathematics Commons^™

Densities For The Repeating Decimals Problems, Nelson A. Carella

Publications and Research

Let \(p\geq 2\) be a prime, and let $1/p=0.\overline{x_{w-1} \ldots x_1x_0}$, $x_i \in \{0,1, 2, \ldots , 9\}$. The period $w\geq 1$ of the repeating decimal $1/p$ is a divisor of $p-1$. This note shows that the counting function for the number of primes with maximal period $w=p-1$ has an effective lower bound $\pi_{10}(x)=\# \{ p\leq x:\ord_p(10)=p-1 \}\gg x/ \log x$. This is a lower bound for the number of primes \(p\leq x\) with a fixed primitive root \(10 \bmod p\) for all large

numbers \(x\geq 1\). An extension to repeating decimal $1/p$ with near maximal period $w=(p-1)/r$, where $r …