Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-11T01:43:13.375Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CONJECTURE OF ERDŐS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  23 February 2012

Adam Tyler Felix  and
M. Ram Murty
Adam Tyler Felix
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany (email: felix@mpim-bonn.mpg.de)
M. Ram Murty
Affiliation:
Department of Mathematics & Statistics, Queen’s University, Jeffery Hall, University Avenue, Kingston, Ontario, Canada K7L 3N6 (email: murty@mast.queensu.ca)
Get access

Abstract

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓa(p)=r}≪εrε for any ε>0, where ℓa(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

MSC classification

Secondary: 11N13: Primes in progressions 11N56: Rate of growth of arithmetic functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 275 - 289
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bombieri, E., Friedlander, J. and Iwaniec, H., Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), 203–251.CrossRefGoogle Scholar
[2]Erdős, P., Bemerkungen zu einer Aufgabe in den Elementen. Arch. Math. (Basel) 27 (1976), 159–163.CrossRefGoogle Scholar
[3]Fomenko, O. M., Class number of indefinite binary quadratic forms and the residual indices of integers modulo p. J. Math. Sci. 122(6) (2004), 3685–3698.CrossRefGoogle Scholar
[4]Goldfeld, D. M., On the number of primes p for which p+a has a large prime factor. Mathematika 16 (1969), 23–27.CrossRefGoogle Scholar
[5]Gupta, R. and Murty, M. R., A remark on Artin’s conjecture. Invent. Math. 78 (1984), 127–130.CrossRefGoogle Scholar
[6]Heath-Brown, D. R., Artin’s conjecture for primitive roots. Q. J. Math. Oxford 37 (1986), 27–38.CrossRefGoogle Scholar
[7]Hooley, C., On Artin’s conjecture. J. Reine Angew. Math. 225 (1967), 209–220.Google Scholar
[8]Hooley, C., Applications of Sieve Methods to the Theory of Numbers, Cambridge University Press (Cambridge, 1976).Google Scholar
[9]Jaeschke, G., Aufgaben: Aufgabe 618. Elem. Math. 26 (1971), 43–44.Google Scholar
[10]Lagarias, J. and Odlyzko, A., Effective versions of the Chebotarev density theorem. In Algebraic Number Fields (ed. Frohlich, A.), Academic Press (New York, 1977), 409–464.Google Scholar
[11]Murty, M. R. and Wong, S., The ABC conjecture and prime divisors of the Lucas and Lehmer sequences. In Number Theory for the Millennium, III (Urbana, IL, 2000), AK Peters (Natick, MA, 2002), 43–54.Google Scholar
[12]Silverman, J., Wieferich’s criterion and the abc-conjecture. J. Number Theory 30 (1988), 226–237.CrossRefGoogle Scholar
[13]Stephens, P. J., Prime divisors of second-order linear recurrences. I. J. Number Theory 8 (1976), 313–332.CrossRefGoogle Scholar
[14]Stewart, C. L., On divisors of Lucas and Lehmer numbers. Preprint, 2011, arXiv:1008.1274[math.NT], pp. 1–18.Google Scholar
[15]Wagstaff, S. S. Jr, Pseudoprimes and a generalization of Artin’s conjecture. Acta Arith. 41 (1982), 141–150.CrossRefGoogle Scholar