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QUADRATIC NONRESIDUES AND NONPRIMITIVE ROOTS SATISFYING A COPRIMALITY CONDITION

Part of: Elementary number theory

Published online by Cambridge University Press:  12 November 2018

JAITRA CHATTOPADHYAY Open the ORCID record for JAITRA CHATTOPADHYAY [Opens in a new window] ,
BIDISHA ROY Open the ORCID record for BIDISHA ROY [Opens in a new window] ,
SUBHA SARKAR Open the ORCID record for SUBHA SARKAR [Opens in a new window]  and
R. THANGADURAI Open the ORCID record for R. THANGADURAI [Opens in a new window]
JAITRA CHATTOPADHYAY
Affiliation:
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India email jaitrachattopadhyay@hri.res.in
BIDISHA ROY*
Affiliation:
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India email bidisharoy@hri.res.in
SUBHA SARKAR
Affiliation:
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India email subhasarkar@hri.res.in
R. THANGADURAI
Affiliation:
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India email thanga@hri.res.in
*
✉bidisharoy@hri.res.in
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Abstract

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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

$$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$
there exists a quadratic nonresidue $g$ which is not a primitive root modulo $p$ such that $\text{gcd}(g,(p-1)/q)=1$.

Keywords

distribution of nonresiduesprimitive rootsfixed point discrete log problem

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 177 - 183
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

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