Results 1 
5 of
5
Prime numbers: a computational perspective. Second Edition
, 2005
"... Cover illustration: The cover shows a magnified view—through a watchmaker’s loupe—of a very small portion of an actual poster giving the 7.2 million decimal digits of the prime 2 240365831. Ifthe entire onemeter tall poster were thus magnified, it would be the size of a baseball diamond. ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Cover illustration: The cover shows a magnified view—through a watchmaker’s loupe—of a very small portion of an actual poster giving the 7.2 million decimal digits of the prime 2 240365831. Ifthe entire onemeter tall poster were thus magnified, it would be the size of a baseball diamond.
A oneparameter quadraticbase version of the Baillie–PSW probable prime test
 Math. Comp
"... Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller test and a “true ” (i.e., with (D/n) =−1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller test and a “true ” (i.e., with (D/n) =−1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1/7710, and pointed out that the lack of counterexamples to the BailliePSW test indicates that the true probability of error may be much lower. In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter: Tu = T mod (T 2 − uT + 1), and define the basecounting functions: B(n) =#{u:0 ≤ u<n, nis a psp(Tu)} and SB(n) =#{u:0 ≤ u<n, nis an spsp(Tu)}. Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n, B(n) <n/2 and SB(n) <n/8, and point out that these are best possible. Finally, based on oneparameter quadraticbase pseudoprimes, we provide a probable prime test, called the OneParameter QuadraticBase Test (OPQBT), which passed by all primes ≥ 5 andpassedbyanoddcompositen = p r1 1 pr2 2 ···prs s (p1 <p2 < ·· · <ps odd primes) with probability of error τ(n). We give explicit formulas to compute τ(n), and prove that
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 for ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 th ..."
Abstract
 Add to MetaCart
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.
NOTES ON SOME NEW KINDS OF PSEUDOPRIMES
"... Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where ..."
Abstract
 Add to MetaCart
Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2pseudoprimes to the same bases. In Section 3, we give examples of Sylow ppseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow ppseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a kfold Carmichael Sylow pseudoprime, ifitisaSylowppseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Sylupsp(b1,...,bk) for short, if it is a Sylppsp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Sylupsp(2, 3, 5)’s < 1016 and some Sylupsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.