Classical Methods of Factorization

  • Hans Riesel
Part of the Modern Birkhäuser Classics book series (MBC)


The art of factoring large integers was not very advanced before the days of the modern computer. Even if there existed some rather advanced algorithms for factorization, invented by some of the most outstanding mathematicians of all times, the amount of computational labor involved discouraged most people from applying those methods to sizable problems. So the field essentially belonged to a few enthusiasts, who however achieved quite impressive results, taking into account the modest means for calculations which they possessed. Famous among these results is F. N. Cole’s factorization in 1903 of 267 − 1 = 193707721 ∙ 761838257287.


Prime Factor Quadratic Residue Search Limit Large Prime Factor Algebraic Factor 
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.


  1. 1.
    R. Sherman Lehman, “Factoring Large Integers,” Math. Comp. 28 (1974) pp. 637–646.MathSciNetMATHCrossRefGoogle Scholar
  2. 1′.
    Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993.MATHGoogle Scholar
  3. 2.
    C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, New Haven, 1966, Art. 329–332.MATHGoogle Scholar
  4. 2′.
    L. G. Sathe, “On a Problem of Hardy on the Distribution of Integers Having a Given Numbers of Prime Factors, I”, Journ. Indian Math. Soc. 17 (1953) pp. 62–82.Google Scholar
  5. 3.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford, 1979, pp. 354–359, 368–370.MATHGoogle Scholar
  6. 4.
    Wladyslaw Narkiewicz, Number Theory, World Scientific, Singapore, 1983, pp. 251– 259.MATHGoogle Scholar
  7. 5.
    Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Math. Monogr. no. 12, John Wiley and Sons, 1959, pp. 74–79.MATHGoogle Scholar
  8. 6.
    Donald E. Knuth and Luis Trabb-Pardo, “Analysis of a Simple Factorization Algorithm,” Theoretical Computer Sc. 3 (1976) pp. 321–348.MathSciNetCrossRefGoogle Scholar
  9. 7.
    Karl Dickman, “On the Frequency of Numbers Containing Prime Factors of a Certain Relative Magnitude” Ark. Mat. Astr. Fys. 22A #10 (1930) pp. 1–14.Google Scholar
  10. 8.
    E.R. Canfield, Paul Erdős and Carl Pomerance, “On a Problem of Oppenheim concerning “Factorisatio Numerorum,” Journ. Number Th. 17 (1983) pp. 1–28.MATHCrossRefGoogle Scholar
  11. 8′.
    J. P. Buhler, H. W. Lenstra, and C. Pomerance, “Factoring Integers with the Number Field Sieve,” in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, New York, 1993, pp. 50–94.CrossRefGoogle Scholar
  12. 9.
    John Brillhart and John L. Selfridge, “Some Factorizations of 2n ± 1 and related results,” Math. Comp. 21 (1967) pp. 87–96.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe Royal Institute of TechnologyStockholmSweden

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