@article{ZT2003,
author={Z. Zhang and M. Tang},
title={Finding strong pseudoprimes to several bases. {II}},
abstract={Define $\psi_m$ to be 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. Thanks to Pomerance et al.\ and Jaeschke, the
$\psi_m$ are known for $1 \leq m \leq 8$. Upper bounds for $\psi_9,\psi_{10}
and \psi_{11}$ were first given by Jaeschke, and those for $\psi_{10}
and \psi_{11}$ were then sharpened by the first author in his
previous paper (Math. Comp. \textbf{70} (2001), 863--872).
In this paper,
we first follow the first author's previous work to use biquadratic residue
characters and cubic residue characters as main tools to tabulate all strong
pseudoprimes (spsp's) $n<10^{24}$ to the first five or six prime bases,
which have the form $n=p q$ with $p, q$ odd primes and $q-1=k(p-1), k=4/3, 5/2, 3/2, 6$;
then we tabulate all Carmichael numbers $<10^{20}$, to the first six prime
bases up to 13, which have the form $n=q_1 q_2 q_3$ with each prime factor
$q_i\equiv 3\mod 4$. There are in total 36 such Carmichael numbers, 12
numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases
17 and 19; one number is an spsp to the first 11 prime bases up to 31.
As a result the upper bounds for $\psi_{9}, \psi_{10}$ and $\psi_{11}$
are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit
number:
$$ \psi_{9}\leq \psi_{10}\leq \psi_{11}\leq Q_{11} &=3825 12305 65464 13051 (19 digits)
= 149491\cdot 747451\cdot 34233211. $$
We conjecture
that $\psi_{9}= \psi_{10}= \psi_{11}=3825\;12305\;65464\;13051,$ and
give reasons to support this conjecture. The main idea for finding these
Carmichael numbers is that we loop on the largest prime factor $q_3$ and
propose necessary conditions on $n$ to be a strong pseudoprime to the first
$5$ prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's,
Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three
prime factors, which are strong pseudoprimes to the first several prime
bases, are given.},
journal= mc,
volume={72},
year={2003},
pages={2085--2097 (electronic)},
number={244},
mrnumber={2004c:11008},
note={\url{http://www.ams.org/journal-getitem?pii=S0025-5718-03-01545-X}}
}
