On the difficulty of finding reliable witnesses

  • W. R. Alford
  • Andrew Granville
  • Carl Pomerance
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 877)


For an odd composite number n, let w(n) denote the least witness for n; that is, the least positive number w for which n is not a strong pseudoprime to the base w. It is widely conjectured, but not proved, that w(n) > 3 for infinitely many n. We show the stronger result that w(n) > (log n)1/(3 log log log n) for infinitely many n. We also show that there are finite sets of odd composites which do not have a reliable witness, namely a common witness for all of the numbers in the set.


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  1. [Ad]
    L. M. Adleman, Two theorems on random polynomial time, Proc. IEEE Symp. Found. Comp. Sci., 19 (1978), 75–83.Google Scholar
  2. [AGP]
    W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Annals Math., to appear.Google Scholar
  3. [ArTANO]
    F. Arnault, Rabin-Miller primality test: composite numbers which pass it, Math. Comp., to appear.Google Scholar
  4. [B]
    E. Bach, Analytic methods in the analysis and design of number-theoretic algorithms, MIT Press, Cambridge, Mass., 1985.Google Scholar
  5. [BH]
    E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp. 61 (1993), 69–82.Google Scholar
  6. [BBCGP]
    P. Beauchemin, G. Brassard, C. Crépeau, C. Goutier and C. Pomerance, The generation of random integers that are probably prime, J. Cryptology 1 (1988), 53–64.Google Scholar
  7. [C]
    M. D. Coleman, On the equation b 1 p − b 2 P 2 = b 3, J. reine angew. Math. 403 (1990), 1–66.Google Scholar
  8. [DLP]
    I. Dåmgard, P. Landrock and C. Pomerance, Average case error estimates for the strong probable prime test, Math. Comp. 61 (1993), 177–194.Google Scholar
  9. [D]
    J. D. Dixon, Factorization and primality tests, Amer. Math. Monthly 91 (1984), 333–352.Google Scholar
  10. [E]
    P. Erdös, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), 201–206.Google Scholar
  11. [F]
    J. B. Friedlander, Shifted primes without large prime factors, in Number Theory and Applications (ed. R. A. Mollin), (Kluwer, NATO ASI, 1989), 393–401.Google Scholar
  12. [HB]
    D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), 265–338.Google Scholar
  13. [HL]
    G. H. Hardy and J. E. Littlewood, Some problems on partitio numerorum III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.Google Scholar
  14. [J]
    G. Jaeschke, On strong pseudoprimes to several bases, Math. Comp. 61 (1993), 915–926.Google Scholar
  15. [Leh]
    D. H. Lehmer, Strong Carmichael numbers, J. Austral. Math. Soc. Ser. A 21 (1976), 508–510.Google Scholar
  16. [Len]
    H. W. Lenstra, Jr., private communication.Google Scholar
  17. [M]
    L. Monier, Evaluation and comparison of two efficient probabilistic primality testing algorithms, Theoret. Comput. Sci. 12 (1980), 97–108.Google Scholar
  18. [P]
    C. Pomerance, On the distribution of pseudoprimes, Math. Comp. 37 (1981), 587–593.Google Scholar
  19. [PSW]
    C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr., The pseudoprimes to 25. 109, Math. Comp. 35 (1980), 1003–1026.Google Scholar
  20. [R]
    M. O. Rabin, Probabilistic algorithm for primality testing, J. Number Theory 12 (1980), 128–138.Google Scholar
  21. [SS]
    R. Solovay and V. Strassen, A fast Monte-Carlo test for primality, SIAM J. Comput. 6 (1977), 84–85; erratum, ibid. 7 (1978), 118.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • W. R. Alford
    • 1
  • Andrew Granville
    • 1
  • Carl Pomerance
    • 1
  1. 1.Department of MathematicsThe University of GeorgiaAthensUSA

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