# On the difficulty of finding reliable witnesses

Conference paper

- First Online:

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

- 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