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


ABSTRACT. Define ψm  to be the smallest strong pseudoprime to all the first  m  prime bases. If we know the exact value of ψm  , we will have, for integers  n < ψm ,  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,  ψm  are known for 1 m 8. Upper bounds for ψ9, ψ10, and ψ11 were given by Jaeschke.

In this paper we tabulate all strong pseudoprimes (spsp's)  n < 1024  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  ψ10  and  ψ11  are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for ψ12 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.