Finding C3-strong pseudoprimes

Let q 1 10 9 and so that ψ 12 = 3186 65857 83403 11511 67461 (24 digits) = 399165290221 . 798330580441, which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those 21978 C 3 -spsp(2)'s is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates N = q 1 q 2 q 3 of C 3 -spsp(2)'s and their prime factors q 1 ,q 2 ,q 3 to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger C 3 -spsp's, say up to 10 50 , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding C 3 -strong pseudoprimes to the first several prime bases are given.