Finding $C_3$-strong pseudoprimes

Let $q_1<q_2<q_3$ be odd primes and $N=q_1q_2q_3$. Put

Then we call $d$ the kernel, the triple $(h_1,h_2,h_3)$ the signature, and $H=h_1h_2h_3$ the height of $N$, respectively. We call $N$ a $C_3$-number if it is a Carmichael number with each prime factor $q_i\equiv 3\mod 4$. If $N$ is a $C_3$-number and a strong pseudoprime to the $t$ bases $b_i$ for $1\leq i\leq t$, we call $N$ a $C_3$-spsp $(b_1, b_2,\dots,b_t)$. Since $C_3$-numbers have probability of error $1/4$ (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of $\psi_m$ (the smallest strong pseudoprime to all the first $m$ prime bases). If we know the exact value of $\psi_m$, we will have, for integers $n<\psi_m$, a deterministic efficient primality testing algorithm which is easy to implement.

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 $<10^{24}$ 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 $<10^{24}$ to the first $12$ prime bases with heights $<10^9$ were found. We conjecture that there exist no $C_3$-spsp's $<10^{24}$to the first $12$ prime bases with heights $\geq 10^9$ and so that

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 $N=q_1q_2q_3$ 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.