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Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
On a combinatorial method for counting smooth numbers in sets of integers
 J. Number Theory
"... In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this r ..."
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In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this result is completely combinatorial and elementary. 1
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
ON THE PRODUCT OF DIVISORS OF n AND OF σ(n)
"... ABSTRACT. For a positive integer n let σ(n) and T (n) be the sum of divisors and product of divisors of n, respectively. In this note, we compare T (n) with T (σ(n)). Key words and phrases: Divisors, Arithmetic functions. 2000 Mathematics Subject Classification. 11A25, 11N56. Let n ≥ 1 be a positive ..."
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ABSTRACT. For a positive integer n let σ(n) and T (n) be the sum of divisors and product of divisors of n, respectively. In this note, we compare T (n) with T (σ(n)). Key words and phrases: Divisors, Arithmetic functions. 2000 Mathematics Subject Classification. 11A25, 11N56. Let n ≥ 1 be a positive integer. In [7], Sándor introduced the function T (n): = ∏ dn d as the multiplicative analog of σ(n), which is the sum of all the positive divisors of n, and studied some of its properties. In particular, he proved several results pertaining to multiplicative perfect numbers, which, by analogy, are numbers n for which the relation T (n) = nk holds with some positive integer k. In this paper, we compare T (n) with T (σ(n)). Our first result is: Theorem 1. The inequality T (σ(n))> T (n) holds for almost all positive integers n. In light of Theorem 1, one can ask whether or not there exist infinitely many n for which T (σ(n)) ≤ T (n) holds. The fact that this is indeed so is contained in the following more precise statement. Theorem 2. Each one of the divisibility relations T (n)  T (σ(n)) and T (σ(n))  T (n) holds for an infinite set of positive integers n. Finally, we ask whether there exist positive integers n> 1 so that T (n) = T (σ(n)). The answer is no. Theorem 3. The equation T (n) = T (σ(n)) has no positive integer solution n> 1. Throughout this paper, for a positive real number x and a positive integer k we write log k x for the recursively defined function given by log k x: = max{log log k−1 x, 1}, where log stands for the natural logarithm function. When k = 1, we simply write log x, and we understand that