Constructing Carmichael Numbers which are Strong Pseudoprimes to Several Bases

We describe here a method of constructing Carmichael numbers which are strong pseudoprimes to some sets of prime bases. We apply it to find composite numbers which are found to be prime by the Rabin-Miller test of packages as Axiom or Maple. We also use a variation of this method to construct strong Lucas pseudoprimes with respect to several pairs of parameters.