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Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
On values taken by the largest prime factor of shifted primes
 Journal of the Australian Mathematical Society
"... Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c( ..."
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Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c(η) p η � has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 ≤ η < (4+3 √ 2)/4, one can take c(η) = 1+ε for any fixed ε> 0. In particular, our results imply that for every real number 0.486 ≤ ϑ ≤ 0.531, the relation P(q − a) ≍ q ϑ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ → P(q − a) for a> 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) 1+o(1) times before it terminates. 1.
ON THE GREATEST PRIME FACTOR OF p − 1 WITH EFFECTIVE CONSTANTS
"... Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there ..."
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Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there is a p<xwith the greatest prime factor of p − 1 exceeding x 3 5. The novelty of our approach is the avoidance of any appeal to Siegel’s Theorem on primes in arithmetic progression. 1.