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QUADRATIC NONRESIDUES AND NONPRIMITIVE ROOTS SATISFYING A COPRIMALITY CONDITION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 2 / April 2019
- Published online by Cambridge University Press:
- 12 November 2018, pp. 177-183
- Print publication:
- April 2019
-
- Article
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- You have access
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Let
$q\geq 1$ be any integer and let 
$\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes 
$p$ satisfying there exists a quadratic nonresidue
$$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$
$g$ which is not a primitive root modulo 
$p$ such that 
$\text{gcd}(g,(p-1)/q)=1$.
