On the difficulty of finding reliable witnesses

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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