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Integers, without large prime factors, in arithmetic progressions, II
"... : 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 Fe ..."
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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 Sunit 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...
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... 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 FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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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 FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
DENSER EGYPTIAN FRACTIONS
, 1998
"... 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 ter ..."
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Cited by 2 (0 self)
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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
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize 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 var ..."
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This paper deals with products of moderatesize 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
The Wild Numbers
, 2004
"... This paper studies the integers that belong the multiplicative semigroup W generated by ..."
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This paper studies the integers that belong the multiplicative semigroup W generated by
Wild and Wooley Numbers
, 2005
"... The wild integer semigroup W(Z) consists of the integers in the multiplicative semigroup generated by { 3n+2 1 ..."
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The wild integer semigroup W(Z) consists of the integers in the multiplicative semigroup generated by { 3n+2 1
THE DISTRIBUTION OF SMOOTH NUMBERS IN ARITHMETIC PROGRESSIONS
, 707
"... We say that a number n is ysmooth if all the prime factors of n lie below y. Let S(y) denote the set of all ysmooth numbers, and let S(x, y) denote the set of ysmooth 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 not ..."
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We say that a number n is ysmooth if all the prime factors of n lie below y. Let S(y) denote the set of all ysmooth numbers, and let S(x, y) denote the set of ysmooth 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