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.

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Abstract. We conjecture that if C is a curve of genus> 1 over a number field k such that C(k) = ∅, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) = ∅. As evidence, we prove a corresponding statement in which C(Fv) is replaced by a random subset of the same size in J(Fv) for each residue field Fv at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Shafarevich-Tate groups are finite, then there exists an algorithm to decide whether a curve over k has a k-point, and the Brauer-Manin obstruction to the Hasse principle for curves over the number fields is the only one. 1. Setup Let k be a number field. Fix an algebraic closure k of k, and let G = Gal(k/k). Let C be a curve of genus g over k. (In this paper, curves are assumed to be smooth, projective, and geometrically integral.) Let C = C ×k k. Let J be the Jacobian of C, which is an abelian variety of dimension g over k. Assume that C has a G-invariant line bundle of degree 1: this

n + 1,..., n + k be consecutive composite numbers. Then for each i, 1 s i s k there is a p i, p i I n + i pi # pi for 1 1 ~ 1 2. 1 2 He also expressed the conjecture in a weaker form stating that any set of k consecutive composite numbers need to have at least k prime factors. We first show that even in this weaker form the conjecture goes far beyond what is known about primes at present. First we define a few number theoretic functions. Denote by o (n, k) the number of distinct prime factors of (n + 1)... (n + k). f 1 (n) is the smallest integer k so that for every 1 s t,4- k v (n, t) Z-t but v (n, k + 1) = k. f0 (n) is the largest integer k for which v (n, k) k k. Clearly f0 (n) i f1 (n) and we shall show that infinitely often f0 (n)> f1 (n). Following Grimm let f2 (n) be the largest integer k so that for each 1-. i!9k there is a p i ln + i, pi 1 1 pi if i 1 / 12. 2 Denote by P(m) the greatest prime factor of m. f.(n) is the