Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:42:53.460Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Remarks on Artin's Conjecture

Published online by Cambridge University Press:  20 November 2018

M. Ram Murty  and
S. Srinivasan
M. Ram Murty
Affiliation:
Department of Mathematics McGill University, Montreal, Canada
S. Srinivasan
Affiliation:
School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect square generates the co-prime residue classes (mod ρ) for infinitely many primes ρ. Let E be the set of a > 1, a not a perfect square, for which Artin's conjecture is false. Set E(x) = card(e ∊ E: e ≤ x). We prove that E(x) = 0(log6 x) and that the number of prime numbers in E is at most 6.

Keywords

10H3510H32Artin's conjectureprimitive roots
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 1 , 01 March 1987 , pp. 80 - 85
Copyright
Copyright © Canadian Mathematical Society 01

References

1. Artin, E., The collected papers of Emil Artin (Lang, S. and Tate, J., Eds.), Reading, Mass., Addison-Wesley 1965; Math. Rev. 31, #1159.Google Scholar
2. Gupta, R. and Ram Murty, M., A remark on Artin's conjecture,Inv. Math. 78(1984) 127–130.Google Scholar
3. Gupta, R., Kumar|Murty, V., and Ram Murty, M., The Euclidean algorithm for S-integers, (to appear).Google Scholar
4. Halberstam, H. and Richert, M., Sieve Methods, Academic Press.Google Scholar
5. Hooley, C., On Artins conjecture, J. Reine Angew. Math. 225(1967) 209–220.Google Scholar
6. Iwaniec, H., Rosser s sieve, Acta Arith. 36(1980) 171–202.Google Scholar