[Zhang 2001a] *Finding strong pseudoprimes to several bases*, Mathematics of Computation, **70**:234 (2001),
863-872.

**A****BSTRACT**. Define *ψ** _{m}* to be the smallest strong pseudoprime to
all the first

In this paper we tabulate
all strong pseudoprimes (spsp's) *n* < 10^{24} to the first ten
prime bases 2, 3, ... , 29, which have the form *n* = *pq** *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 *ψ*_{10}_{ }_{ }and_{ }*ψ*_{11 }are lowered from 28-
and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for *ψ*_{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.