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

 

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 efficient  primality  testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm  are known for 1 ≤ m ≤ 8. Upper bounds for ψ9, ψ10, and ψ11  were first given by Jaeschke, and those for ψ10, and ψ11   were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

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 < 1024   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 < 1020, to the first six prime bases up to 13, which have the form  n = q1 q2 q3    with each prime factor qi ≡ 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  ψ9, ψ10, and ψ11  are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

ψ9  ≤ ψ10  ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051

= 149491  ·  747451  ·  34233211.

We conjecture that

ψ9  = ψ10  =  ψ11 = 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 q3 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.