Distribution of Irreducible Polynomials over F2

  • Kenneth H. Hicks
  • Gary L. Mullen
  • Ikuro Sato


Using a polynomial analogue of the wheel sieve, we discuss the distribution of irreducible polynomials over F 2. In particular, we provide considerable numerical evidence that in analogue to integer arithmetic progressions, irreducible polynomials over F 2 are binomially distributed in the progressions of the wheel sieve. We also present, numerical evidence that the irreducibles of fixed degree are binomially distributed by weight. Also briefly discussed is the distribution of self-reciprocal irreducible polynomials. A number of conjectures are raised.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I.F. Blake, S. Gao, R.J. Lambert, Construction and distribution problems for irreducible trinomials over finite fields, Applications of Finite Fields Edited by D. Gollmann, Clarendon Press, Oxford, 1996, 19–32.Google Scholar
  2. 2.
    D.R. Hayes, The distribution of irreducibles in GF[q,x], Trans. Amer. Math. Soc. 117 (1965), 101–127.MathSciNetzbMATHGoogle Scholar
  3. 3.
    K.H. Hicks AND I. Sato, Heuristics of arithmetic progressions in the framework of the wheel sieve,submitted for publication.Google Scholar
  4. 4.
    D. Jungnickel, Finite Fields: Structure and Arithmetics Bibliographisches Inst. & F.A. Brockhaus AG, Mannheim, 1993.Google Scholar
  5. 5.
    H. Kornblum, Uber die Primfunktionen in einer arithmetischen Progression, Math. Z. 5 (1919), 100–111.MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Lidl and H. Niederreiter, Finite Fields Cambridge Univ. Press, 1997.Google Scholar
  7. 7.
    P. Pritchard, Explaining the wheel sieve, Acta Informat. 17 (1982), 477–485.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Kenneth H. Hicks
    • 1
  • Gary L. Mullen
    • 2
  • Ikuro Sato
    • 1
  1. 1.Department of PhysicsOhio UniversityAthensUSA
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations