Reference Database (references for the Prime Pages) Home Search Site Largest Finding How Many? Mersenne Glossary Prime Curios! e-mail list FAQ Prime Lists Titans Submit primes This is the Prime Pages' interface to our BibTeX database.  Rather than being an exhaustive database, it just lists the references we cite on these pages.  Please let me know of any errors you notice.References: [ Home | Author index | Key index | Search ] Zhang2000 Z. Zhang, "Finding strong pseudoprimes to several bases," Math. Comp., 70:234 (2001) 863--872.  MR 2001g:11009 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=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 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. Another prime page by Chris K. Caldwell