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Large primes and Fermat factors
 Math. Comp
, 1998
"... Abstract. A systematic search for large primes has yielded the largest Fermat factors known. ..."
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Abstract. A systematic search for large primes has yielded the largest Fermat factors known.
Sieve with Two Large Primes
"... This paper deals with variations of the Quadratic Sieve integer factoring algorithm. We describe what we believe is the rst implementation of the Hypercube Multiple Polynomial Quadratic Sieve with two large primes, We have used this program to factor many integers with up to 116 digits. Our program ..."
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This paper deals with variations of the Quadratic Sieve integer factoring algorithm. We describe what we believe is the rst implementation of the Hypercube Multiple Polynomial Quadratic Sieve with two large primes, We have used this program to factor many integers with up to 116 digits. Our
NFS with Four Large Primes: An Explosive Experiment
, 1995
"... The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multilarge prime variation of the general number field sieve integer factoring algorithm (NFS, cf. [8]). For traditional factoring algorithms that make use of at most two large primes, the ..."
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The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multilarge prime variation of the general number field sieve integer factoring algorithm (NFS, cf. [8]). For traditional factoring algorithms that make use of at most two large primes
On the asymptotic distribution of large prime factors
 J. London Math. Soc
, 1993
"... A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly reordering the comp ..."
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A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly re
Sieving the positive integers by large primes
, 1988
"... Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous ..."
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Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous
The ThreeLargePrimes Variant of the Number Field Sieve
"... The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this ..."
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of this method (but fortunately, also the easiest to parallelise). Pollard's original method allowed one large prime. After that the twolargeprimes variant led to substantial improvements [11]. In this paper we investigate whether the threelargeprimes variant may lead to any further improvement. We
On ideals free of large prime factors
 DE BORDEAUX 16 (2004), 733–772
, 2004
"... In 1989, E. Saias established an asymptotic formula for Ψ(x, y) = {n ≤ x: p  n ⇒ p ≤ y}  with a very good error term, valid for exp ( (log log x) (5/3)+ɛ) ≤ y ≤ x, x ≥ x0(ɛ), ɛ> 0. We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous func ..."
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In 1989, E. Saias established an asymptotic formula for Ψ(x, y) = {n ≤ x: p  n ⇒ p ≤ y}  with a very good error term, valid for exp ( (log log x) (5/3)+ɛ) ≤ y ≤ x, x ≥ x0(ɛ), ɛ> 0. We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous function ΨK(x, y) with the same error term and valid in the same region. Our main objective is to compare the formulae for Ψ(x, y) and ΨK(x, y), and in particular to compare the second term in the two expansions.
DISTRIBUTION OF SPECIAL SEQUENCES MODULO A LARGE Prime
, 2003
"... We study the sets {g x −g y (modp):1 ≤ x, y ≤ N} and {xy:1 ≤ x, y ≤ N} where p is a large prime number, g is a primitive root, and p 2/3 <N<p. ..."
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We study the sets {g x −g y (modp):1 ≤ x, y ≤ N} and {xy:1 ≤ x, y ≤ N} where p is a large prime number, g is a primitive root, and p 2/3 <N<p.
On strings of consecutive integers with no large prime factors
 J. Austral. Math. Soc. Ser. A
, 1998
"... We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ", we show that there are infinitely many strings of consecutive integers of size about n, free of prime ..."
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We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ", we show that there are infinitely many strings of consecutive integers of size about n, free of prime
Results 1  10
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