On Tuesday, July 15, 2003, at 11:22 AM, Schroeppel, Richard wrote:
There is a small prize offered for a number that is a pseudoprime base 2 and also a Lucas PSP. Probably the first examples will have many prime factors. The Carmichael numbers will usually be PSP base 2 and 3. There are an infinite number of them. I don't know the situation if we strengthen the requirement to MR-PSP. Richard Pinch has produced a list of PSP2s < 10^16; this would be a good starting point for trying out tests.
I've just been looking through a copy of Daniel Bleichenbacher's dissertation (ETH, 1996), which is available from his homepage at http://www.bell-labs.com/user/bleichen/ Also there is a list of the numbers which are strong pseudoprimes to both bases 2 and 3 up to 10^16. It has been checked that Mma's Lucas test catches all of these, though apparently Mma version 2.2 used a weaker Lucas test and reported that n = 89 * 11551 * 37159 was PrimeQ (Pinch '93, as reported in B's dissertation). --Michael Kleber kleber@brandeis.edu