Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T14:25:06.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudoprime reductions of elliptic curves

Published online by Cambridge University Press:  01 May 2009

ALINA CARMEN COJOCARU ,
FLORIAN LUCA  and
IGOR E. SHPARLINSKI
ALINA CARMEN COJOCARU
Affiliation:
Dept. of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, U.S.A. and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania. e-mail: cojocaru@math.uic.edu
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México. e-mail: fluca@matmor.unam.mx
IGOR E. SHPARLINSKI
Affiliation:
Dept. of Computing, Macquarie University, Sydney, NSW 2109, Australia. e-mail: igor@ics.mq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Let b ≥ 2 be an integer and let E/ be a fixed elliptic curve. In this paper, we estimate the number of primes p ≤ x such that the number of points nE(p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of nE(p).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 513 - 522
Copyright
Copyright © Cambridge Philosophical Society 2008

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

REFERENCES

[BaKaWi]Bassily, N. L., Kátai, I. and Wijsmuller, M.. On the prime power divisors of the iterates of the Euler-ϕ function. Publ. Math. Debrecen 55 (1999), 17–32.CrossRefGoogle Scholar
[CrPo]Crandall, R. and Pomerance, C.. Prime Numbers: A Computational Perspective. (Springer-Verlag, 2005).Google Scholar
[Co]Cojocaru, A. C.. Reductions of an elliptic curve with almost prime orders. Acta Arith. 119 (2005), 265–289.CrossRefGoogle Scholar
[CoFoMu]Cojocaru, A. C., Fouvry, É. and Murty, M. R.. The square sieve and the Lang–Trotter conjecture. Canadian J. Math. 57 (2005), 1155–1177.CrossRefGoogle Scholar
[FoIw]Fouvry, É. and Iwaniec, H.. Gaussian primes. Acta Arith. 79 (1997), 249–287.CrossRefGoogle Scholar
[HaWr]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. (Oxford University Press, 1979).Google Scholar
[Iv]Ivić, A.. The Riemann Zeta-Function. Theory and Applications. (Dover Publications, Inc., 2003).Google Scholar
[LaOd]Lagarias, J. C. and Odlyzko, A. M.. Effective versions of the Chebotarev density theorem. Algebraic Number Fields (Academic Press, 1977), 409–464.Google Scholar
[LuSh1]Luca, F. and Shparlinski, I. E.. Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function. Indag. Math. 17 (2006), 611–625.CrossRefGoogle Scholar
[LuSh2]Luca, F. and Shparlinski, I. E.. Pseudoprime Cullen and Woodall numbers. Colloq. Math. 107 (2007), 35–43.CrossRefGoogle Scholar
[PoRo]van der Poorten, A. J. and Rotkiewicz, A.. On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc. Ser. A 29 (1980), 316–321.CrossRefGoogle Scholar