Moree, Pieter; Sha, Min Primes in arithmetic progressions and nonprimitive roots. (English) Zbl 1472.11031 Bull. Aust. Math. Soc. 100, No. 3, 388-394 (2019). Summary: 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. Cited in 2 Documents MSC: 11A15 Power residues, reciprocity 11N13 Primes in congruence classes 11N69 Distribution of integers in special residue classes Keywords:primitive root; near-primitive root; Artin’s primitive root conjecture; arithmetic progression Software:OEIS PDFBibTeX XMLCite \textit{P. Moree} and \textit{M. Sha}, Bull. Aust. Math. Soc. 100, No. 3, 388--394 (2019; Zbl 1472.11031) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Genocchi irregular primes. References: [1] On-line encyclopedia of integer sequences, ‘Genocchi irregular primes’, sequence A321217, available at https://oeis.org. [2] R.Gupta and M. R.Murty, ‘A remark on Artin’s conjecture’, Invent. Math.78 (1984), 127-130.10.1007/BF01388719 · Zbl 0549.10037 [3] D. R.Heath-Brown, ‘Artin’s conjecture for primitive roots’, Q. J. Math. (2)37 (1986), 27-38.10.1093/qmath/37.1.27 · Zbl 0586.10025 [4] C.Hooley, ‘On Artin’s conjecture’, J. reine angew. Math225 (1967), 209-220. · Zbl 0221.10048 [5] S.Hu and M.-S.Kim, ‘The (S, {2})-Iwasawa theory’, J. Number Theory158 (2016), 73-89.10.1016/j.jnt.2015.06.013 · Zbl 1400.11141 [6] S.Hu, M.-S.Kim, P.Moree and M.Sha, ‘Irregular primes with respect to Genocchi numbers and Artin’s primitive root conjecture’, J. Number Theory (to appear), arXiv:1809.08431. · Zbl 1473.11051 [7] H. W.LenstraJr., ‘On Artin’s conjecture and Euclid’s algorithm in global fields’, Invent. Math.42 (1977), 202-224.10.1007/BF01389788 · Zbl 0362.12012 [8] H. W.LenstraJr., P.Moree and P.Stevenhagen, ‘Character sums for primitive root densities’, Math. Proc. Cambridge Philos. Soc.157 (2014), 489-511.10.1017/S0305004114000450 · Zbl 1353.11104 [9] P.Moree, ‘On primes in arithmetic progression having a prescribed primitive root’, J. Number Theory78 (1999), 85-98. · Zbl 0931.11036 [10] P.Moree, ‘On the distribution of the order and index of g (mod p) over residue classes I’, J. Number Theory114 (2005), 238-271.10.1016/j.jnt.2004.09.004 · Zbl 1099.11052 [11] P.Moree, ‘On primes in arithmetic progression having a prescribed primitive root II’, Funct. Approx. Comment. Math.39 (2008), 133-144.10.7169/facm/1229696559 · Zbl 1223.11118 [12] P.Moree, ‘Artin’s primitive root conjecture—a survey’, Integers12A (2012), Article ID A13, 100 pages. · Zbl 1271.11002 [13] P.Moree, ‘Near-primitive roots’, Funct. Approx. Comment. Math.48 (2013), 133-145.10.7169/facm/2013.48.1.11 · Zbl 1300.11006 [14] J.-P.Serre, ‘Quelques applications du théorème de densité de Chebotarev’, Publ. Math. Inst. Hautes Études Sci.54 (1981), 323-401. [15] S. S.WagstaffJr., ‘Pseudoprimes and a generalization of Artin’s conjecture’, Acta Arith.41 (1982), 141-150.10.4064/aa-41-2-141-150 · Zbl 0496.10001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.