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The Mathematical Intelligencer

, Volume 27, Issue 2, pp 26–34 | Cite as

Integers and polynomials: comparing the close cousins Z and F q [x]

  • Gove Effinger
  • Kenneth Hicks
  • Gary L. Mullen
Article

Keywords

Prime Number Finite Field Irreducible Polynomial Monic Polynomial Riemann Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    L. M. Adleman and M.-D.A. Huang, Primality Testing and Abelian Varieties over Finite Fields, Lect. Notes in Math., Vol. 1512, Springer-Verlag, 1992.Google Scholar
  2. [2]
    M. Agrawal, N. Kayal, and N. Saxena,PRIMES is in P, Annals of Mathematics160 (2004),s 781–798.MATHMathSciNetGoogle Scholar
  3. [3]
    Z. I. Borevich and I. R. Shafarevich,Number Theory, New York, Academic Press, 1966.Google Scholar
  4. [4]
    V. Brun,La serie 1/5 + 1/7 + 1/11 +. . .oú les dénominateurs sont “nombres premiers jumeaux“ est convergente ou finie, Bull. Sci. Math.43 (1919), 100–104 and 124–128.MATHGoogle Scholar
  5. [5]
    J. R. Chen,On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II, Acta Math. Sci. Sinica16 (1973), 157–176; and 21 (1978), 421–430.MATHGoogle Scholar
  6. [6]
    J. R. Chen and Y. Wang,On the Odd Goldbach Problem, Acta Math. Sci. Sinica32 (1989), 702–718.MATHGoogle Scholar
  7. [7]
    R. Crandall and C. Pomerance,Prime Numbers: A Computational Perspective, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
  8. [8]
    J.-M. Deshouillers, G. Effinger, H. nte Riele, and D. Zinoviev,A Complete Vinogradov 3-Primes Theorem under the Riemann Hypothesis, Electronic Research Announcements of the AMS,3 (1997), 99–104.MATHGoogle Scholar
  9. [9]
    G. Effinger,A Goldbach Theorem for Polynomials of Low Degree over Odd Finite Fields, Acta Arithmetica,42 (1983), 329–365.MATHMathSciNetGoogle Scholar
  10. [10]
    G. Effinger,A Goldbach 3-Primes Theorem for Polynomials of Low Degree over Finite Fields of Characteristic 2, Journal of Number Theory, 29 (1988), 345–363.MATHMathSciNetGoogle Scholar
  11. [11]
    G. Effinger,The Polynomial 3-Primes Conjecture, inComputer Assisted Analysis and Modeling on the IBM 3090, Baldwin Press, Athens, Georgia 1992.Google Scholar
  12. [12]
    G. Effinger and D.R. Hayes, Additive Number Theory of Polynomials over Finite Fields,Oxford University Press, 1991.Google Scholar
  13. [13]
    G. Effinger, K. Hicks, and G.L. Mullen,Twin Irreducible Polynomials over Finite Fields, in: Finite Fields with Applications in Coding Theory, Cryptography, and Related Areas, Springer, Heidelberg, 2002, 94–111.Google Scholar
  14. [14]
    J.A. Gallian,Contemporary Abstract Algebra, Houghton-Mifflin, 2002.Google Scholar
  15. [15]
    S. Gao,On the Deterministic Complexity of Polynomial Factoring, J. Symbolic Computation,31 (2001), 19–36.MATHGoogle Scholar
  16. [16]
    J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge Univ. Press, 1999.Google Scholar
  17. [17]
    H. Halberstan and H.E. Richert,Sieve Methods, Academic Press, New York, 1974.Google Scholar
  18. [18]
    G.H. Hardy and J.E. Littlewood,Some Problems of ‘Partitio Numerorum’; III: On the Expression of a Number as a Sum of Primes, Acta Mathematics44 (1922), 1–70.MathSciNetGoogle Scholar
  19. [19]
    G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Clarendon Press, Oxford, 1959.Google Scholar
  20. [20]
    D.R. Hayes,A Goldbach Theorem for Polynomials with Integer Coefficients, Amer. Math. Monthly,72 (1965), 45–46.MATHMathSciNetGoogle Scholar
  21. [21]
    D.R. Hayes,The Expression of a Polynomial As a Sum of Three Inducibles, Acta Arithmetica,11 (1966), 461–488.MATHMathSciNetGoogle Scholar
  22. [22]
    A.K. Lenstra and H.W. Lenstra, Jr, editors,The Development of the Number Field Sieve, Lect. Notes in Math.,1554, Springer-Verlag, 1993.Google Scholar
  23. [23]
    R. Lidl and H. Niederreiter,Finite Fields, Cambridge Univ. Press, 1997.Google Scholar
  24. [24]
    H. Niederreiter,A New Efficient Factorization Algorithm for Polynomials over Small Finite Fields, Appl. Alg. in Eng., Comm., and Comp. 4(1993), 81–87.MATHMathSciNetGoogle Scholar
  25. [25]
    T.R. Nicely,Enumeration to 1014 of the Twin Primes and Brun’s Constant, Virginia Journal of Science46 (1995) 195–204. Enumeration to 1.6 x 1016 is available from web sitehttp://www.trnicely.net. MathSciNetGoogle Scholar
  26. [26]
    A.M. Odlyzko, M. Rubinstein, and M. Wolf,Jumping Champions, Exp. Math. 8(1999) 107–118.MATHMathSciNetGoogle Scholar
  27. [27]
    I. Peterson,Cranking Out Primes, Science News, Oct. 1, 1994, 217.Google Scholar
  28. [28]
    G. Polya,Heuristic Reasoning in the Theory of Numbers, Am. Math. Monthly,66 (1959), 375–384.MATHMathSciNetGoogle Scholar
  29. [29]
    P. Ribenboim,The New Book of Prime Numbers, Springer-Verlag, 1995.Google Scholar
  30. [30]
    K. Scheicher and J.M. Thuswaldner,Digit Systems in Polynomial Rings over Finite Fields, Finite Fields Appl. 9 (2003), 322–333.MATHMathSciNetGoogle Scholar
  31. [31]
    A. Selberg,Reflections around Ramanujan Centenary, Paper 41 inCollected Papers/Atle Selberg, Vol. 1, Springer-Verlag, Berlin, 1989.Google Scholar
  32. [32]
    P.W. Shor,Polynomial-time Algorithms for Primie Factorization and Discrete Logarithms on a Quantum Computer, SIAM J. Comp. 26 (1997), 1484–1509.MATHMathSciNetGoogle Scholar
  33. [33]
    J. Teitelbaum, Review ofPrime Numbers: A Computational Perspective, Bull. Amer. Math. Soc,39 (2002), 449–454.Google Scholar
  34. [34]
    R.C. Vaughan,The Hardy-LittlewoodMethod (2nd ed.), Cambridge University Press, 1997.Google Scholar
  35. [35]
    I.M. Vinogradov,Representation of an odd number as a sum of three primes, Comptes Rendus (Doklady) de I’Academia des Sciences de I’USSR15 (1937), 291–294.Google Scholar
  36. [36]
    A. Weil,Sur les courbes algébriques et les variétés qui s’en déduisent, Hermann, 1948.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsSkidmore CollegeSaratoga SpringsUSA
  2. 2.Department of Physics and AstronomyOhio UniversityAthensUSA
  3. 3.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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