@article{Zhang2000,
author={Z. Zhang},
title={Finding strong pseudoprimes to several bases},
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 primality testing algorithm which
is not only easier to implement but also faster than either the Jacobi
sum test or the elliptic curve test. Thanks to Pomerance et al.\ and Jaeschke,
$\psi_m$ are known for $1 \leq m \leq 8$. Upper bounds for $\psi_9,\psi_{10}$
and $\psi_{11}$ were given by Jaeschke. In this paper we tabulate all strong
pseudoprimes (spsp's) $n<10^{24}$ to the first ten prime bases $2, 3, \cdots,
29,$ which have the form $n=p,q$ with $p, q$ odd primes and $q-1=k(p-1),
k=2, 3, 4.$ There are in total 44 such numbers, six of which are also spsp(31),
and three numbers are spsp's to both bases 31 and 37. As a result the upper
bounds for $\psi_{10}$ and $\psi_{11}$ are lowered from 28- and 29-decimal-digit
numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound
for $\psi_{12}$ is obtained. The main tools used in our methods are the
biquadratic residue characters and cubic residue characters. We propose
necessary conditions for $n$ to be a strong pseudoprime to one or to several
prime bases. Comparisons of effectiveness with both Jaeschke's and Arnault's
methods are given.},
journal= mc,
volume= 70,
year= 2001,
pages={863--872},
number={234},
mrnumber={2001g:11009}
}
