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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 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.
A VARIANT OF THE BOMBIERIVINOGRADOV THEOREM WITH EXPLICIT CONSTANTS AND APPLICATIONS
"... ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of t ..."
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ABSTRACT. We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λ and a Dirichlet character χ. As a consequence of this result we prove an effective variant of the BombieriVinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes. For integers a and q ≥ 1, let 1.