On $q$-orders in primitive modular groups
Volume 166 / 2014
Acta Arithmetica 166 (2014), 397-404
MSC: 11R45, 11M41, 11M06, 11Z05.
DOI: 10.4064/aa166-4-5
Abstract
We prove an upper bound for the number of primes $p \leq x$ in an arithmetic progression $1 \pmod Q$ that are exceptional in the sense that $\mathbb{Z}^*_p$ has no generator in the interval $[1, B].$ As a consequence we prove that if $Q >\exp \bigl[c\frac{\log p}{\log B} (\log \log p)\big]$ with a sufficiently large absolute constant $c$, then there exists a prime $q$ dividing $Q$ such that $\nu_q(\mathop{\rm ord}_p b) =\nu_q(p-1)$ for some positive integer $b\le B.$ Moreover we estimate the number of such $q$'s under suitable conditions.
