TITLE: Do we really know the reliability of PrimeQ[n] (for $n>10^{16}$)?
QUESTION [39 upvotes]: The algorithm Mathematica uses for its PrimeQ function is described on MathWorld. That web page says PrimeQ uses, "the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test." It also says, 

"As of 1997, this procedure is known to be correct only for all
  $n<10^{16}$, but no counterexamples are known and if any exist, they
  are expected to occur with extremely small probability (i.e., much
  less than the probability of a hardware error on a computer performing
  the test)."

More details on the algorithm are given in A Course in Computational Number Theory by David Bressoud. That book says there are 52,593 integers less than $10^{16}$ that are 2- and 3- strong pseudoprimes. It also suggests Lucas pseudoprimes less than $10^{16}$ are rare. Number theory is a very rigorous field. Has any rigorous work been done to substantiate the claim above that, if a PrimeQ counterexamples exist, they are expected to occur with extremely small probability. All I can find so far is an extrapolation of the trend for ($n<10^{16}$), and that is a very hand-wavy argument.
Besides that, I am aware of no efforts to extend the brute force testing beyond $10^{16}$. With Moore's law and 15 years of work, considerable progress could have been made in that way.

REPLY [4 votes]: Regarding the original question: it is not necessarily the case that Lucas pseudoprimes less than 1E16 are rare.  The point is that, if you make a list of psp's base 2 and a list of Lucas psp's, then the two lists are observed to have very little, if any, overlap.
As for probabilities: In the Miller-Rabin test, if n is composite, then 3/4 of the bases are witnesses for the compositeness of n.
For the strong Lucas test, the corresponding probability (measuring (P,Q) pairs), is 11/15; see F. Arnault, "The Rabin-Monier Theorem for Lucas Pseudoprimes". Mathematics of Computation 66 (218) (April, 1997), pp. 869–881.