[Zhang
2005] *Finding C _{3 }*-

**A****BSTRACT**. Let *q*_{1} <_{ }*q*_{2} <* **q*_{3} be odd primes
and *N** *= *q*_{1}* **q*_{2}* **q*_{3}. Put

*d* = gcd(*q*_{1}−1, *q*_{2}−1,* **q*_{3}−1) and *h** _{i}* = (

Then
we call *d* the* kernel*, the triple (*h*_{1}, *h*_{2},* **h*_{3}) the *signature*,
and *H *= *h*_{1}*h*_{2}*h*_{3} the *height*
of *N*, respectively. We call *N* a *C*_{3}-number if it is a
Carmichael number with each prime factor *q** _{i}* ≡ 3 mod 4.
If

In this paper, we first describe an algorithm for finding *C*_{3}-spsp(2)'s, to a given limit, with heights
bounded. There are in total 21978 *C*_{3}-spsp(2) 's
with heights < 10^{9} . We then give an overview of the 21978 *C*_{3}-spsp(2)'s and tabulate 54 of
them, which are *C*_{3}-spsp's to the first 8 prime bases up to 19; three numbers are spsp's
to the first 11 prime bases up to 31. No *C*_{3}-spsp's to the first 12 prime bases with heights <10^{9} were found. We conjecture that
there exist no *C*_{3}-spsp's to the first 12 prime bases with heights
≥ 10^{9} and so that

*ψ*_{12 =}3186 65857 83403 11511 67461

= 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 -strong pseudoprimes
to the first several prime bases are given.