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Proceedings of the American Mathematical Society

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Explicit upper bounds on the least primitive root


Authors: Kevin J. McGown and Tim Trudgian
Journal: Proc. Amer. Math. Soc. 148 (2020), 1049-1061
MSC (2010): Primary 11A07, 11L40
DOI: https://doi.org/10.1090/proc/14800
Published electronically: October 28, 2019
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Abstract: We give a method for producing explicit bounds on $ g(p)$, the least primitive root modulo $ p$. Using our method we show that

$\displaystyle g(p)<2r\,2^{r\omega (p-1)}\,p^{\frac {1}{4}+\frac {1}{4r}}$

for $ p>10^{56}$ where $ r\geq 2$ is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of $ p$. For example, our method also allows us to show that $ g(p)<p^{5/8}$ for all $ p\geq 10^{22}$ and $ g(p)<p^{1/2}$ for $ p\geq 10^{56}$.

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Additional Information

Kevin J. McGown
Affiliation: Department of Mathematics and Statistics, California State University, Chico, California 95929; and School of Science, P.O. Box 7916, The University of New South Wales, Canberra, ACT, 2610 Australia
Email: kmcgown@csuchico.edu

Tim Trudgian
Affiliation: School of Science, P.O. BOX 7916, The University of New South Wales, Canberra, ACT, 2610 Australia
Email: t.trudgian@adfa.edu.au

DOI: https://doi.org/10.1090/proc/14800
Received by editor(s): April 28, 2019
Received by editor(s) in revised form: July 24, 2019
Published electronically: October 28, 2019
Additional Notes: The second author was supported by Australian Research Council Future Fellowship FT160100094
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 American Mathematical Society