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Divisibility

  • W. A. Coppel
Chapter
Part of the Universitext book series (UTX)

Abstract

In the set ℕ of all positive integers we can perform two basic operations: addition and multiplication. In this chapter we will be primarily concerned with the second operation.

Keywords

Integral Domain Division Ring Great Common Divisor Irreducible Polynomial Primitive Root 
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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Selected References

  1. [1]
    W.R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. Math. 139 (1994), 703–722.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S.A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955), 361–386.MATHMathSciNetGoogle Scholar
  3. [3]
    D.A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Math. 83 (1994), 327–330.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Davenport, The higher arithmetic, 7th ed., Cambridge University Press, 1999.Google Scholar
  5. [5]
    L.E. Dickson, History of the theory of numbers, 3 vols., Carnegie Institute, Washington, D.C., 1919–1923. [Reprinted, Chelsea, New York, 1992.]Google Scholar
  6. [6]
    P.G.L. Dirichlet, Lectures on number theory, with supplements by R. Dedekind, English transl. by J. Stillwell, American Mathematical Society, Providence, R.I., 1999. [German original, 1894.]MATHGoogle Scholar
  7. [7]
    J.D. Dixon, Factorization and primality tests, Amer. Math. Monthly 91 (1984), 333–352.MATHCrossRefGoogle Scholar
  8. [8]
    D.W. Dubois and A. Steger, A note on division algorithms in imaginary quadratic fields, Canad. J. Math. 10 (1958), 285–286.MATHMathSciNetGoogle Scholar
  9. [9]
    R.B. Eggleton, C.B. Lacampagne and J.L. Selfridge, Euclidean quadratic fields, Amer. Math. Monthly 99 (1992), 829–837.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    W.J. Ellison, Waring's problem, Amer. Math. Monthly 78 (1971), 10–36.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Euclid, The thirteen books of Euclid's elements, English translation by T.L. Heath, 2nd ed., reprinted in 3 vols., Dover, New York, 1956.Google Scholar
  12. [12]
    S. Gao, Absolute irreducibility of polynomials via Newton polytopes, J. Algebra 237 (2001), 501–520.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    C.F. Gauss, Disquisitiones arithmeticae, English translation by A.A. Clarke, revised by W.C. Waterhouse, Springer, New York, 1986. [Latin original, 1801.]Google Scholar
  14. [14]
    E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York, 1985.MATHGoogle Scholar
  15. [15]
    G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, 2008.Google Scholar
  16. [16]
    D.R. Heath-Brown, Artin's conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), 27–38.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A.P. Hillman and V.E. Hoggatt, Exponents of primes in generalized binomial coefficients, J. Reine Angew. Math. 262/3 (1973), 375–380.MathSciNetGoogle Scholar
  18. [18]
    L.K. Hua, Introduction to number theory, English translation by P. Shiu, Springer-Verlag, Berlin, 1982.Google Scholar
  19. [19]
    A. Hurwitz, Über die Zahlentheorie der Quaternionen, Mathematische Werke, Band II, pp. 303–330, Birkhäuser, Basel, 1933.Google Scholar
  20. [20]
    N. Iiyori and H. Yamaki, On a conjecture of Frobenius, Bull. Amer. Math. Soc. (N.S) 25 (1991), 413–416.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    N. Jacobson, Basic Algebra I, 2nd ed., W.H. Freeman, New York, 1985.MATHGoogle Scholar
  22. [22]
    J.-R. Joly, Équations et variétés algébriques sur un corps fini, Enseign. Math. (2) 19 (1973), 1–117.MATHMathSciNetGoogle Scholar
  23. [23]
    S. Lang, Algebra, corrected reprint of 3rd ed., Addison-Wesley, Reading, Mass., 1994.Google Scholar
  24. [24]
    D.H. Lehmer, Guide to tables in the theory of numbers, National Academy of Sciences, Washington, D.C., reprinted 1961.Google Scholar
  25. [25]
    D.N. Lehmer, List of prime numbers from 1 to 10,006,721, reprinted, Hafner, New York, 1956.Google Scholar
  26. [26]
    D.N. Lehmer, Factor table for the first ten millions, reprinted, Hafner, New York, 1956.Google Scholar
  27. [27]
    A.K. Lenstra, Primality testing, Proc. Symp. Appl. Math. 42 (1990), 13–25.MathSciNetGoogle Scholar
  28. [28]
    W.J. Le Veque, Fundamentals of number theory, reprinted Dover, Mineola, N.Y., 1996.Google Scholar
  29. [29]
    U. Libbrecht, Chinese mathematics in the thirteenth century, MIT Press, Cambridge, Mass., 1973.MATHGoogle Scholar
  30. [30]
    R. Lidl and H. Niederreiter, Finite fields, 2nd ed., Cambridge University Press, 1997.Google Scholar
  31. [31]
    H.B. Mann, Introduction to algebraic number theory, Ohio State University, Columbus, Ohio, 1955.MATHGoogle Scholar
  32. [32]
    R. Narasimhan, Complex analysis in one variable, Birkhäuser, Boston, Mass., 1985.MATHGoogle Scholar
  33. [33]
    W. Narkiewicz, Number theory, English translation by S. Kanemitsu, World Scientific, Singapore, 1983.MATHGoogle Scholar
  34. [34]
    I. Niven, H.S. Zuckerman and H.L. Montgomery, An introduction to the theory of numbers, 5th ed., Wiley, New York, 1991.Google Scholar
  35. [35]
    H. Prüfer, Untersuchungen über Teilbarkeitseigenschaften, J. Reine Angew. Math. 168 (1932), 1–36.Google Scholar
  36. [36]
    T.-S. Rhai, A characterization of polynomial domains over a field, Amer. Math. Monthly 69 (1962), 984–986.CrossRefMathSciNetGoogle Scholar
  37. [37]
    R.L. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 (1978), 120–126.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    P. Salmon, Sulla fattorialità delle algebre graduate e degli anelli locali, Rend. Sem. Mat. Univ. Padova 41 (1968), 119–138.MATHMathSciNetGoogle Scholar
  39. [39]
    P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945–952.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282–301.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    A. Scholz, Einführung in die Zahlentheorie, revised and edited by B. Schoeneberg, 5th ed., de Gruyter, Berlin, 1973.Google Scholar
  42. [42]
    H.J.S. Smith, Report on the theory of numbers, Collected mathematical papers, Vol. 1, pp. 38–364, reprinted, Chelsea, New York, 1965. [Original, 1859–1865.]Google Scholar
  43. [43]
    T.J. Stieltjes, Sur la théorie des nombres, Ann. Fac. Sci. Toulouse 4 (1890), 1–103. [Reprinted in Tome 2, pp. 265–377 of T.J. Stieltjes, Oeuvres complètes, 2 vols., Noordhoff, Groningen, 1914–1918.]Google Scholar
  44. [44]
    R.C. Vaughan, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997.MATHCrossRefGoogle Scholar
  45. [45]
    E. Waring, Meditationes algebraicae, English transl. of 1782 edition by D. Weeks, Amer. Math. Soc., Providence, R.I., 1991.Google Scholar
  46. [46]
    A. Weil, Number theory: an approach through history, Birkhäuser, Boston, Mass., 1984.MATHGoogle Scholar
  47. [47]
    A.E. Western and J.C.P. Miller, Tables of indices and primitive roots, Royal Soc. Math. Tables, Vol. 9, Cambridge University Press, London, 1968.Google Scholar
  48. [48]
    H.C. Williams, Primality testing on a computer, Ars Combin. 5 (1978), 127–185.MATHMathSciNetGoogle Scholar

Additional References

  1. M. Agarwal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. 160 (2004), 781–793. [An unconditional deterministic polynomial-time algorithm for determining if an integer > 1 is prime or composite.]CrossRefMathSciNetGoogle Scholar
  2. A. Granville, It is easy to determine whether a given integer is prime, Bull. Amer. Math. Soc. (N.S.) 42 (2005), 3–38.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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