Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψt, we will have, for integers n<ψt, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψt are known for 1 ≤ t ≤ 8. Conjectured values of ψ9,...,ψ12 were given by us in our previous papers (Math. Comp. 72 (2003), 2085–2097; 74 (2005), 1009–1024). The main purpose of this paper is to give exact values of ψ ′ t for 13 ≤ t ≤ 19; to give a lower bound of ψ ′ 20: ψ ′ 20> 1036; and to give reasons and numerical evidence of K2- and C3-spsp’s < 1036 to support the following conjecture: ψt = ψ ′ t <ψ′ ′ t for any t ≥ 12, where ψ ′ t (resp. ψ′ ′ t) is the smallest K2- (resp. C3-) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp’s < 1036 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p−1); and that a C3-spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1.