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PRIMES IN ARITHMETIC PROGRESSIONS AND NONPRIMITIVE ROOTS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  24 May 2019

PIETER MOREE Open the ORCID record for PIETER MOREE [Opens in a new window]  and
MIN SHA Open the ORCID record for MIN SHA [Opens in a new window]
PIETER MOREE
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany email [email protected]
MIN SHA*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia email [email protected]
*
✉[email protected]
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Abstract

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$-near primitive root and we prove a more difficult variant.

Keywords

primitive rootnear-primitive rootArtin’s primitive root conjecturearithmetic progression

MSC classification

Primary: 11N13: Primes in progressions
Secondary: 11N69: Distribution of integers in special residue classes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 388 - 394
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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References

On-line encyclopedia of integer sequences, ‘Genocchi irregular primes’, sequence A321217, available at https://oeis.org.Google Scholar
Gupta, R. and Murty, M. R., ‘A remark on Artin’s conjecture’, Invent. Math. 78 (1984), 127–130.10.1007/BF01388719Google Scholar
Heath-Brown, D. R., ‘Artin’s conjecture for primitive roots’, Q. J. Math. (2) 37 (1986), 27–38.10.1093/qmath/37.1.27Google Scholar
Hooley, C., ‘On Artin’s conjecture’, J. reine angew. Math 225 (1967), 209–220.Google Scholar
Hu, S. and Kim, M.-S., ‘The (S, {2})-Iwasawa theory’, J. Number Theory 158 (2016), 73–89.10.1016/j.jnt.2015.06.013Google Scholar
Hu, S., Kim, M.-S., Moree, P. and Sha, M., ‘Irregular primes with respect to Genocchi numbers and Artin’s primitive root conjecture’, J. Number Theory (to appear), arXiv:1809.08431.Google Scholar
Lenstra, H. W. Jr., ‘On Artin’s conjecture and Euclid’s algorithm in global fields’, Invent. Math. 42 (1977), 202–224.10.1007/BF01389788Google Scholar
Lenstra, H. W. Jr., Moree, P. and Stevenhagen, P., ‘Character sums for primitive root densities’, Math. Proc. Cambridge Philos. Soc. 157 (2014), 489–511.10.1017/S0305004114000450Google Scholar
Moree, P., ‘On primes in arithmetic progression having a prescribed primitive root’, J. Number Theory 78 (1999), 85–98.Google Scholar
Moree, P., ‘On the distribution of the order and index of g (mod p) over residue classes I’, J. Number Theory 114 (2005), 238–271.10.1016/j.jnt.2004.09.004Google Scholar
Moree, P., ‘On primes in arithmetic progression having a prescribed primitive root II’, Funct. Approx. Comment. Math. 39 (2008), 133–144.10.7169/facm/1229696559Google Scholar
Moree, P., ‘Artin’s primitive root conjecture—a survey’, Integers 12A (2012), Article ID A13, 100 pages.Google Scholar
Moree, P., ‘Near-primitive roots’, Funct. Approx. Comment. Math. 48 (2013), 133–145.10.7169/facm/2013.48.1.11Google Scholar
Serre, J.-P., ‘Quelques applications du théorème de densité de Chebotarev’, Publ. Math. Inst. Hautes Études Sci. 54 (1981), 323–401.Google Scholar
Wagstaff, S. S. Jr., ‘Pseudoprimes and a generalization of Artin’s conjecture’, Acta Arith. 41 (1982), 141–150.10.4064/aa-41-2-141-150Google Scholar