Finding strong pseudoprimes to several bases
WebFeb 1, 2010 · In this paper we tabulate all strong pseudoprimes (spsp’s) n<10 24 to the first ten prime bases 2,3,⋯,29, which have the form n=pq with p,q odd primes and q-1=k(p-1), k=2,3,4· There are in... WebApr 1, 2001 · In this paper we tabulate all strong pseudoprimes (spsp’s) n<10 24 to the first ten prime bases 2,3,⋯,29, which have the form n=pq with p,q odd primes and q-1=k (p-1), k=2,3,4· There are in...
Finding strong pseudoprimes to several bases
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WebThe Baillie–PSW test is a combination of a strong Fermat probable prime test (that means Miller-Rabin) to base 2 and a strong Lucas probable prime test. The Fermat and Lucas test each have their own list of pseudoprimes, that is, composite numbers that pass the test. For example, the first ten strong pseudoprimes to base 2 are WebApr 26, 2024 · The test with a random integer a is called strong pseudoprime test to base a. If the test does not say the number n is composite then n is called a strong pseudoprime to base a. The set of all strong pseudoprimes to base a is denoted by spsp ( a ).
WebMar 24, 2024 · A strong pseudoprime to the base is also an Euler pseudoprime to the base (Pomerance et al. 1980). The strong pseudoprimes include some Euler … WebOct 1, 2003 · 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 …
WebIf n and 2 n − 1 are primes , where n > 3 , then p = n ⋅ ( 2 n − 1) is a pseudoprime to base 3 . Claim: If p is an odd prime not dividing a 2 − 1, then m := a 2 p − 1 a 2 − 1 is a pseudoprime base a. Proof: We know a p ≡ a mod p, so we also know a 2 p ≡ a 2 mod p. So we know that p ∣ ( a 2 p − a 2). WebSep 2, 2015 · We also present an algorithm to find all integers that are strong pseudoprimes to the first prime bases; with a reasonable heuristic assumption we can …
WebAt last we speed our algorithm for finding larger $C_3$-spsp’s, say up to $10^ {50}$, with a given signature to more prime bases. Comparisons of effectiveness with Arnault’s and our previous methods for finding $C_3$-strong pseudoprimes to the first several prime bases are given. References Similar Articles Additional Information
WebMar 24, 2024 · A composite number passing Miller's test is called a strong pseudoprime to base . If a number does not pass the test, then it is called a witness for the compositeness. If is an odd, positive composite number, then passes Miller's test for at most bases with (Long 1995). There is no analog of Carmichael numbers for strong pseudoprimes . cheater spy app freeWebSep 16, 2011 · Euler Pseudoprimes for Half of the Bases Authors: Lorenzo Di Biagio Italian National Institute of Statistics Abstract We prove that an odd number is an Euler pseudoprime for exactly one half of... cyclohexanehexone octahydrateWebFeb 17, 2000 · FINDING STRONG PSEUDOPRIMES TO SEVERAL BASES ZHENXIANG ZHANG Dedicated to the memory of P. Erd}os (1913{1996) Abstract. De ne mto be the smallest strong pseudoprime to all the rst mprime bases. If we know the exact value of … cyclohexane hs codeWebadshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A cyclohexane hnmr signalsWebDefine 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 cyclohexane h nmrWebJun 1, 2024 · The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality... cheater spy shopWebIn this paper we tabulate all strong pseudoprimes (spsp’s) n<1024 to the first ten prime bases 2, 3, ·· · , 29, which have the form n = pq with p, q odd primes and q −1 =k(p −1),k=2, 3, 4. There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp’s to both bases 31 and 37. cyclohexane hexyl