Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T03:28:14.906Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Counting Function of Elliptic Carmichael Numbers

Published online by Cambridge University Press:  20 November 2018

Florian Luca  and
Igor E. Shparlinski
Florian Luca
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, Máxico e-mail: fluca@matmor.unam.mx
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia e-mail: igor.shparlinski@mq.edu.au
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.

We give an upper bound for the number of elliptic Carmichael numbers $n\,\le \,x$ that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non $\text{CM}$). We also discuss several possible further improvements.

Keywords

11Y1111N36elliptic Carmichael numbersapplications of sieve methods
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 105 - 112
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Cranfield, E. R., Erdős, P., and Pomerance, C., On a problem of Oppenheim concerning “factorisationumerorum”. J. Number Theory 17 (1983), no. 1, 1–28. http://dx.doi.org/10.1016/0022-314X(83)90002-1 Google Scholar
[2] Cojocaru, A. C., Fouvry, É., and Murty, M. R., The square sieve and the Lang–Trotter conjecture. Canad. J. Math. 57 (2005), no. 6, 1155–1177. http://dx.doi.org/10.4153/CJM-2005-045-7 Google Scholar
[3] Cojocaru, A. C., Luca, F., and Shparlinski, I. E., Pseudoprime reductions of elliptic curves. Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 513–522. http://dx.doi.org/10.1017/S0305004108001758 Google Scholar
[4] David, C. and Wu, J., Pseudoprime reductions of elliptic curves. Canad. J. Math. 64 (2012), no. 1, 81–101. http://dx.doi.org/10.4153/CJM-2011-044-x Google Scholar
[5] Schoof, R., The exponents of the group of points on the reduction of an elliptic curve. In: Arithmetic algebraic geometry (Texel, 1989), Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991, pp. 325–335.Google Scholar
[6] Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259–331. http://dx.doi.org/10.1007/BF01405086 Google Scholar
[7] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151, Springer-Verlag, Berlin, 1995.Google Scholar
[8] Silverman, J. H., Elliptic Carmichael numbers and elliptic Korselt criteria. arxiv:1108.3830. Google Scholar
[9] Tenenbaum, G., Introduction to analytic and probabilistic number theory. Cambridge Studies in Mathematics, 46, Cambridge University Press, Cambridge, 1995.Google Scholar