[Zhang
and Tang 2003]
(with Min Tang) *Finding strong pseudoprimes to several bases.** II*,
Mathematics of Computation, **72**:244 (2003), 2085-2097.

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

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* = *pq** * 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}* ≡ 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

*ψ*_{9 }≤ *ψ*_{10 }≤ *ψ*_{11 }≤ *Q*_{11 }= 3825
12305 65464 13051

=
149491 · 747451 · 34233211._{}

We conjecture that

*ψ*_{9 }= *ψ*_{10 }= *ψ*_{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.