The Recognition of Primes

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


One very important concern in number theory is to establish whether a given number N is prime or composite. At first sight it might seem that in order to decide the question an attempt must be made to factorize N and if it fails, then N is a prime. Fortunately there exist primality tests which do not rely upon factorization. This is very lucky indeed, since all factorization methods developed so far are rather laborious. Such an approach would admit only numbers of moderate size to be examined and the situation for deciding on primality would be rather bad. It is interesting to note that methods to determine primality, other than attempting to factorize, do not give any indication of the factors of N in the case where N turns out to be composite. — Since the prime 2 possesses certain particular properties, we shall, in this and the next chapter, assume for most of the time that N is an odd integer.


Primitive Root Primality Test Fermat Number Composite Number Compositeness Test 
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.
    D. H. Lehmer, “On the Converse of Fermat’s Theorem,” Amer.Math. Monthly 43 (1936) pp. 347–354.MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. H. Lehmer, “On the Converse of Fermat’s Theorem II,” Amer. Math. Monthly 56 (1949) pp. 300–309.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Carl Pomerance, John L. Selfridge and Samuel S. Wagstaff Jr., “The Pseudoprimes to 25 • 109,” Math. Comp. 35 (1980) pp. 1003–1026.MathSciNetMATHGoogle Scholar
  4. 4.
    Oystein Ore, Number Theory and Its History, McGraw-Hill, New York, 1948, pp. 331–339.MATHGoogle Scholar
  5. 5.
    Gary Miller, “Riemann’s Hypothesis and Tests for Primality,” Journ. of Comp. and Syst. Sc. 13 (1976) pp. 300–317.MATHCrossRefGoogle Scholar
  6. 6.
    Carl Pomerance, “On the Distribution of Pseudoprimes,” Math. Comp. 37 (1981) pp. 587–593.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    John Brillhart, D. H. Lehmer and John Selfridge, “New Primality Criteria and Factorizations of 2 ± 1,” Math. Comp. 29 (1975) pp. 620–647.MathSciNetMATHGoogle Scholar
  8. 8.
    Daniel Shanks, “Corrigendum,” Math. Comp. 39 (1982) p. 759.MathSciNetGoogle Scholar
  9. 9.
    H. C. Williams and J. S. Judd, “Some Algorithms for Prime Testing Using Generalized Lehmer Functions,” Math. Comp. 30 (1976) pp. 867–886.MathSciNetMATHGoogle Scholar
  10. 10.
    Hans Riesel, “Lucasian Criteria for the Primality of N = h • 2n — 1,” Math. Comp. 23 (1969), pp. 869–875.MathSciNetMATHGoogle Scholar
  11. 11.
    K. Inkeri and J. Sirkesalo, “Factorization of Certain Numbers of the Form h • 2n + k,” Ann. Univ. Turkuensis, Series A No. 38 (1959)Google Scholar
  12. 12.
    K. Inkeri,“Tests for Primality,” Ann. Acad. Sc. Fenn., Series A No. 279 (1960)Google Scholar
  13. 13.
    William Adams and Daniel Shanks, “Strong Primality Tests That Are Not Sufficient,” Math. Comp. 39 (1982) pp. 255–300.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Leonard Adleman and Frank T. Leighton, “An O(n 1/10.89) Primality Testing Algorithm,” Math. Comp.36 (1981) pp. 261–266.MathSciNetMATHGoogle Scholar
  15. 15.
    Carl Pomerance, “Recent Developments in Primality Testing,” The Mathematical Intelligencer 3 (1981) pp. 97–105.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Carl Pomerance, “The Search for Prime Numbers,” Sc. Amer. 247 (Dec. 1982) pp. 122–130.CrossRefGoogle Scholar
  17. 17.
    Leonard M. Adleman, Carl Pomerance and Robert S. Rumely, “On Distinguishing Prime Numbers from Composite Numbers,” Ann. of Math. 117 (1983) pp. 173–206.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H. W. Lenstra Jr., “Primality Testing Algorithms,” Séminaire Bourbaki 33 (1980–81) No. 576, pp. 243–257.Google Scholar
  19. 19.
    H. Cohen and H. W. Lenstra Jr., “Primality Testing and Jacobi Sums,” Math. Comp. 42 (1984) pp. 297–330.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    John D. Dixon, “Factorization and Primality Tests,” Am. Math. Monthly 91 (1984) pp. 333–352. Contains a large bibliography.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Hans Riesel
    • 1
  1. 1.VällingbySweden

Personalised recommendations