Abstract
Let p be an odd prime. If an integer g generates a subgroup of index t in \(({\mathbb {Z}}/p{\mathbb {Z}})^{*}\), then we say that g is a t-near primitive root modulo p. In this paper, for a subset \(\{ a_{1}, a_{2}, \dots , a_{n} \}\) of \({\mathbb {Z}} \setminus \{-1, 0, 1\}\), we prove each coprime residue class contains a positive density of primes p not having \(a_{i}\) as a t-near primitive root and with the \(a_{i}\) satisfying a prescribed residue pattern modulo p, for \(1 \le i \le n\). We also prove a more refined variant of it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balasubramanian R, Luca F and Thangadurai R, On the exact degree of \({\mathbb{Q}}(\sqrt{a_{1},}\sqrt{a_{2}},\dots ,\sqrt{a_{\ell }})\) over \(\mathbb{Q}\), Proc. Amer. Math. Soc. 138(7) (2010) 2283–2288
Dey P K and Kumar B, An analogue of Artin’s primitive root conjecture, Integers, 16 (2016) Paper No. A67, 7
Dummit D S and Foote R M, Abstract Algebra, third edition (2004) (Hoboken, NJ: John Wiley & Sons)
Fried M, Arithmetical properties of value sets of polynomials, Acta Arith. 15 (1968/69) 91–115
Hooley C, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967) 209–220
Karthick Babu C G and Mukhopadhyay A, Quadratic residue pattern and the Galois group of \({\mathbb{Q}}(\sqrt{a_{1},}\sqrt{a_{2}},\dots ,\sqrt{a_{\ell }})\) over \({\mathbb{Q}}\), Proc. Amer. Math. Soc. 150(10) (2022) 4277–4285
Karthick Babu C G, Mukhopadhyay A and Sahu S, On the explicit galois group of \({\mathbb{Q}} (\sqrt{a_{1}}, \sqrt{a_{2}}, \dots , \sqrt{a_{n}}, \zeta _{d})\) over \(\mathbb{Q}\), arxiv:2201.01159-v1 (2022)
Lenstra Jr. H W, Moree P and Stevenhagen P, Character sums for primitive root densities, Math. Proc. Cambridge Philos. Soc. 157(3) (2014) 489–511
Matthews K R, A generalisation of Artin’s conjecture for primitive roots, Acta Arith. 29(2) (1976) 113–146
Moree P, On primes in arithmetic progression having a prescribed primitive root, J. Number Theory 78(1) (1999) 85–98
Moree P, On the distribution of the order and index of \(g~({\rm mod} \; p)\) over residue classes. I, J. Number Theory 114(2) (2005) 238–271
Moree P, On primes in arithmetic progression having a prescribed primitive root. II, Funct. Approx. Comment. Math. 39(Part 1) (2008) 133–144
Moree P, Near-primitive roots, Funct. Approx. Comment. Math. 48(Part 1) (2013) 133–145
Moree P and Sha M, Primes in arithmetic progressions and nonprimitive roots, Bull. Aust. Math. Soc. 100(3) (2019) 388–394
Wagstaff Jr. S S, Pseudoprimes and a generalization of Artin’s conjecture, Acta Arith. 41(2) (1982) 141–150
Weintraub S H, Galois Theory, second edition (2009) (New York: Springer, Universitext)
Wright S, Patterns of quadratic residues and nonresidues for infinitely many primes, J. Number Theory 123(1) (2007) 120–132
Wright S, Quadratic residues and the combinatorics of sign multiplication, J. Number Theory 128(4) (2008) 918–925
Acknowledgements
The authors would like to express their sincere thanks to Prof. Pieter Moree for helpful suggestions in improving this paper. They are also thankful to the anonymous referee for some fruitful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B Sury.
Rights and permissions
About this article
Cite this article
Karthick Babu, C.G., Sahu, S. Non primitive roots with a prescribed residue pattern. Proc Math Sci 133, 9 (2023). https://doi.org/10.1007/s12044-023-00728-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-023-00728-4
Keywords
- Primes in congruence classes
- quadratic residue
- quadratic extensions
- cyclotomic extensions
- near-primitive roots