Finding strong pseudoprimes to several bases. II

Abstract

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 ψ 9 , ψ 10 and ψ 11 were first given by Jaeschke, and those for ψ 10 and ψ 11 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 < 10 24 to the first five or six prime bases, which have the form n = p q 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 and ψ 11 are lowered from 20-and 22-decimal-digit numbers to a 19-decimal-digit number: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.

Strong Pseudo Primes to Base 2

“…The algorithm starts with an integer n which passes the strong pseudo prime test to base 2. Several studies have been conducted to give an explicit characterization of such composite integers [7,14,20,21]. The experimental results show that a majority of such integers n have only two factors.…”

“…A partial answer, which gives all elements of spsp less than a certain integer, is also presented in it. The idea later expanded in many ways [7,20,21]. For example, suppose one finds the set S of all composite numbers less than t which are spsp to prime bases p 1 , .…”

Strong pseudoprimes to the first eight prime bases

“…In paper [1], Jaeschke also gave upper bounds for ψ 9 , ψ 10 , ψ 11 . These bounds were improved by Z. Zhang for several times and finally he conjectured that ψ 9 = ψ 10 = ψ 11 = Q 11 = 3825 12305 65464 13051 = 149491 · 747451 · 34233211 Zhang also gave upper bounds and conjectures for ψ m , with 12 ≤ m ≤ 20 (see [4,5,6]).…”