Given Primes p,q, Find Pseudoprime pqr

Suppose we're looking for a general way with arbitrary primes p,q 
to find a prime r such that pqr is a pseudoprime to one of a small 
set of bases.  In principle this is just a finite search.  Taking the 
base 2 as an example, we know 2^r = 2 (mod r) for every prime r, so 
the condition that 2^(pqr) = 2 (mod pqr) implies that 2^(pq) = 2 
(mod r).  Therefore, r must be a divisor of 2^(pq-1) - 1.

For example, with p=3 and q=5 we need r to be a divisor of
2^14 - 1 = (3)(43)(127).  Therefore, the only possible values of
r that could make pqr a base-2 pseudoprime are 3, 43, and 127.  
It turns out that both 43 and 127 give base-2 pseudoprimes 
(645 and 1905).

For another example, take p=3 and q=11.  In this case r must be
a divisor of 2^32 - 1 = (3)(5)(17)(257)(65537).  It turns out that
r=257 gives the base-2 pseudoprime 8481.

Of course, we can't really make use of this to test for the primality 
of n=pq by finding r, because finding the factors of 2^(n-1) - 1 is 
more or less the same as knowing the factorization of n-1, and if we 
know that, we can already test for the primality of n using standard 
methods.