Strong Pseudoprimes to Twelve Prime Bases
نویسندگان
چکیده
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.
منابع مشابه
Some new kinds of pseudoprimes
We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. It turns out that every n < 1012, which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow p-pseudoprime to two of these bases for an appropriate prime p|n− 1. We also give examples of strong pseudoprimes to many bases...
متن کاملNotes on some new kinds of pseudoprimes
J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime 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 unfavora...
متن کاملFinding strong pseudoprimes to several bases II
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, ...
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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Math. Comput.
دوره 86 شماره
صفحات -
تاریخ انتشار 2017