Let N be an integer with at least two distinct prime factors. We reduce the problem of factoring N to the task of finding t + 2 integer solutions (e1 ; : : : ; e t ) 2 ZZ t of the inequalities fi fi fi fi fi t X i=1 e i log p i \Gamma log N fi fi fi fi fi N \Gammac p o(1) t t X i=1 je i log p i j (2c \Gamma 1) log N + 2 log p t ; where c ? 1 is fixed and p1 ; : : : ; p t are the first t primes. We show, under a reasonable hypothesis, that there are N "+o(1) many solutions (e1 ; : : : ; e t ) where " = c\Gamma1\Gamma(2c\Gamma1)=ff with p t = (log N) ff . Here we have " ? 0 if and only if ff ? (2c \Gamma 1)=(c \Gamma 1). We associate with the primes p1 ; : : : ; p t a lattice L ae IR t+1 of rank t and we associate with N a point N 2 IR t+1 . The above problem of diophantine approximation amounts to finding lattice vectors z that are sufficiently close to N in the 1--norm. We also reduce the problem of computing, for a prime N , discrete logarithms of the units...

Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.

Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there is no irreducible trinomial of degree r.

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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-ordering the components of A(«), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite-dimensional simplex of vectors (xv x2,...) having non-negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(/i) to the corresponding Poisson-Dirichlet distribution on this simplex; this result was obtained by Billingsley [3]. 1.

: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of S--unit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...

This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFT-based power-series exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.