On the difficulty of finding reliable witnesses

  • W. R. Alford
  • Andrew Granville
  • Carl Pomerance
Conference paper

DOI: 10.1007/3-540-58691-1_36

Part of the Lecture Notes in Computer Science book series (LNCS, volume 877)
Cite this paper as:
Alford W.R., Granville A., Pomerance C. (1994) On the difficulty of finding reliable witnesses. In: Adleman L.M., Huang MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg

Abstract

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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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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