Building Carmichael numbers with a large number of prime factors and generalization to other numbers

We extend the method of Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes. In particular, we exhibit the first known strong Fibonacci pseudoprimes. The method can be viewed as a simplified version, yet practical, of the method used by Alford, Granville and Pomerance to prove that there is an infinite number of Carmichael numbers