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Finding strong pseudoprimes to several bases. II,Math
 Department of Mathematics, Anhui Normal University
"... 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 ..."
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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 = q1q2q3 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 22decimaldigit numbers to a 19decimaldigit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 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. 1.
TWO KINDS OF STRONG PSEUDOPRIMES UP TO 10 36
"... 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 ..."
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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 C3spsp’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 K2spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p−1); and that a C3spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1.
Article electronically published on November 2, 2004 FINDING C3STRONG PSEUDOPRIMES
"... Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael numberwitheachprimefactorqi≡3 ..."
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Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael numberwitheachprimefactorqi≡3 mod 4. If N is a C3number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, wecallNaC3spsp(b1,b2,...,bt). Since C3numbers have probability of error 1/4 (the upper bound of that for the RabinMiller test), they often serve as the exact values or upper bounds of ψm (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. In this paper, we first describe an algorithm for finding C3spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3spsp(2)’s < 1024 with heights < 109. We then give an overview of the 21978 C3spsp(2)’s and tabulate 54 of them, which are C3spsp’s to the first 8 prime bases up to 19; three numbers are spsp’s to the first 11 prime bases up to 31. No C3spsp’s < 1024 to the first 12 prime bases with heights < 109 were found. We conjecture that there exist no C3spsp’s < 1024 to the first 12 prime bases with heights ≥ 109 and so that