• Hans Riesel
Part of the Progress in Mathematics book series (PM, volume 57)


The art of decomposing large integers into prime factors has advanced considerably during the last 15 years. It is the advent of high-speed computers that has rekindled interest in this field. This development has followed several lines. In one of these, already existing theoretical methods and known algorithms have been carefully analyzed and perfected. As an example of this work we mention Michael Morrison and John Brillhart’s analysis of an old factorization method, the continued fraction algorithm, going back to ideas introduced already by Legendre and developed further by Maurice Kraitchik, D. H. Lehmer and R. E. Powers.


Prime Factor Factorization Method Quadratic Residue Continue Fraction Expansion Search Limit 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Sherman Lehman, “Factoring Large Integers,” Math. Comp. 28 (1974) pp. 637–646.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, New Haven, 1966, Art. 329–332.MATHGoogle Scholar
  3. 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
  4. 4.
    Wladyslaw Narkiewicz, Number Theory, World Scientific, Singapore, 1983, pp. 251–259.MATHGoogle Scholar
  5. 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
  6. 6.
    Donald E. Knuth and Luis Trabb-Pardo, “Analysis of a Simple Factorization Algorithm,” Theoretical Computer Sc. 3 (1976) pp. 321–348.MathSciNetCrossRefGoogle Scholar
  7. 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
  8. 8.
    J. M. Pollard, “Theorems on Factorization and Primality Testing,” Proc. Cambr. Philos. Soc. 76 (1974) pp. 521–528.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    H. C. Williams, “A p + 1 Method of Factoring,” Math. Comp. 39 (1982) pp. 225–234.MathSciNetMATHGoogle Scholar
  10. 10.
    J. M. Pollard, “A Monte Carlo Method for Factorization,” Nordisk Tidskrift ör Informationsbehandling (BIT) 15 (1975) pp. 331–334.MathSciNetMATHGoogle Scholar
  11. 11.
    Richard P. Brent, “An Improved Monte Carlo Factorization Algorithm,” Nordisk Tidskrift für Informationsbehandling (BIT) 20 (1980) pp. 176–184.MathSciNetMATHGoogle Scholar
  12. 12.
    Richard P. Brent and J. M. Pollard, “Factorization of the Eighth Fermat Number,” Math. Comp. 36 (1981) pp. 627–630.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    John Brillhart and John L. Selfridge, “Some Factorizations of 2’ ± 1 and Related Results,” Math. Comp. 21 (1967) pp. 87–96.MathSciNetMATHGoogle Scholar
  14. 14.
    Daniel Shanks, “Class Number, A Theory of Factorization, and Genera,” Amer. Math. Soc. Proc. Symposia in Pure Math. 20 (1971) pp. 415–440.MathSciNetGoogle Scholar
  15. 15.
    Louis Monier, Algorithmes de Factorisations d’Entiers, IRIA, Paris, 1980, pp. 3.13–3.24.Google Scholar
  16. 16.
    R. J. Schoof, “Quadratic Fields and Factorization,” printed in H. W. Lenstra, Jr. and R. Tijdeman, Computational Methods in Number Theory, Part II, Mathematisch Centrum, Amsterdam 1982, pp. 235–286.Google Scholar
  17. 17.
    Michael A. Morrison and John Brillhart, “A Method of Factoring and the Factorization of F 7,” Math. Comp. 29 (1975) pp. 183–205.MathSciNetMATHGoogle Scholar
  18. 18.
    D. H. Lehmer and R. E. Powers, “On Factoring Large Numbers,” Bull. Am. Math. Soc. 37 (1931) pp. 770–776.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maurice Kraïtchik, Théorie des Nombres. Tome II, Gauthiers-Villars, Paris, 1926, pp. 195–208.MATHGoogle Scholar
  20. 20.
    Carl Pomerance, “Analysis and Comparison of Some Integer Factoring Algorithms,” printed in H. W. Lenstra, Jr. and R. Tijdeman, Computational Methods in Number Theory, Part I, Mathematisch Centrum Tract 154, Amsterdam 1982, pp. 89–139.Google Scholar
  21. 21.
    Carl Pomerance and Samuel S. Wagstaff, Jr., “Implementation of the Continued Fraction Integer Factoring Algorithm,” Congr. Num. 37 (1983) pp. 99–118.MathSciNetGoogle Scholar
  22. 22.
    Thorkil Naur, Integer Factorization, DAIMI report, University of Aarhus, 1982.Google Scholar
  23. 23.
    A. A. Mullin, “Recursive Function Theory,” Bull. Am. Math. Soc. 69 (1963) p. 737.CrossRefGoogle Scholar
  24. 24.
    James A. Davis and Diane B. Holdridge, “Factorization Using the Quadratic Sieve Algorithm, ”SANDIA report, SAND83–1346, SANDIA National Laboratories, Livermore, 1983.Google Scholar
  25. 25.
    John D. Dixon, “Asymptotically Fast Factorization of Integers,” Math. Comp. 36 (1981) pp. 255–260.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    C. P. Schnorr and H. W. Lenstra, Jr., “A Monte Carlo Factoring Algorithm with Linear Storage,” Math. Comp. 43 (1984) pp. 289–311.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Dennis Parkinson and Marvin Wunderlich, “A Compact Algorithm for Gaussian Elimination over GF(2) Implemented on Highly Parallel Computers,” Parallel Computing 1 (1984) pp. 65–73.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Hans Riesel
    • 1
  1. 1.VällingbySweden

Personalised recommendations