Strong pseudoprimes to the first eight prime bases
نویسندگان
چکیده
منابع مشابه
Strong Pseudoprimes to Twelve Prime Bases
Let ψm be the smallest strong pseudoprime to the first m prime bases. This value is known for 1 ≤ m ≤ 11. We extend this by finding ψ12 and ψ13. We also present an algorithm to find all integers n ≤ B that are strong pseudoprimes to the first m prime bases; with a reasonable heuristic assumption we can show that it takes at most B2/3+o(1) time.
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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...
متن کاملFinding strong pseudoprimes to several bases
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 primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9, ...
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Building upon the work of Carl Pomerance and others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function [8]. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up ...
متن کاملFinding C3-strong pseudoprimes
Let q1 < q2 < q3 be odd primes and N = q1q2q3. Put d = gcd(q1 − 1, q2 − 1, q3 − 1) and hi = qi−1 d , i = 1, 2, 3. Then we call d the kernel, the triple (h1, h2, h3) the signature, and H = h1h2h3 the height of N , respectively. We call N a C3-number if it is a Carmichael number with each prime factor qi ≡ 3 mod 4. If N is a C3-number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, we c...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2014
ISSN: 0025-5718,1088-6842
DOI: 10.1090/s0025-5718-2014-02830-5