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: 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.

This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1

This paper studies the integers that belong the multiplicative semigroup W generated by

The wild integer semigroup W(Z) consists of the integers in the multiplicative semigroup generated by { 3n+2 1

An Egyptian fraction is a sum of reciprocals of distinct positive integers, so called because the ancient Egyptians represented rational numbers in that way. In an earlier paper, the author [8] showed that every positive rational number r has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator. More

We say that a number n is y-smooth if all the prime factors of n lie below y. Let S(y) denote the set of all y-smooth numbers, and let S(x, y) denote the set of y-smooth numbers below x. Let Ψ(x, y) denote the number of smooth integers below x; thus Ψ(x, y) is the cardinality of S(x, y). In this note we consider the distribution of smooth numbers among