Abstract. A systematic search for large primes has yielded the largest Fermat factors known.

This paper deals with variations of the Quadratic Sieve integer factoring algorithm. We describe what we believe is the rst imple-mentation 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

The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multi-large prime varia-tion 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

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

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 y-t cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous

of this method (but fortunately, also the easiest to parallelise). Pollard's original method allowed one large prime. After that the two-large-primes variant led to substantial improvements [11]. In this paper we investigate whether the three-large-primes variant may lead to any further improvement. We

Abstract not found

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.

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.

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