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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
On Carmichael numbers in arithmetic progressions
 J. Aust. Math. Soc
"... We dedicate this paper to our friend Alf van der Poorten Assuming a weak version of a conjecture of HeathBrown on the least prime in a residue class, we show that for any coprime integers a and m> 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. 1 1 ..."
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We dedicate this paper to our friend Alf van der Poorten Assuming a weak version of a conjecture of HeathBrown on the least prime in a residue class, we show that for any coprime integers a and m> 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. 1 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
THERE ARE INFINITELY MANY PERRIN PSEUDOPRIMES
"... Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zeroden ..."
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Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zerodensity estimates for Hecke Lfunctions. 1. Background In a 1982 paper [1], Adams and Shanks introduced a probable primality test based on third order recurrence sequences. The following is a version of that test. Consider sequences An = An(r, s) defined by the following relations: A−1 = s, A0 = 3, A1 = r, and An = rAn−1 − sAn−2 + An−3. Let f(x) = x 3 − rx 2 + sx − 1 be the associated polynomial and ∆ its discriminant. (Perrin’s sequence is An(0, −1).) Definition. The signature mod m of an integer n with respect to the sequence Ak(r, s) is the 6tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an Ssignature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Qsignature if its signature mod n is congruent to (A, s, B, B, r, C), where for some integer a with f(a) ≡ 0 mod n, A ≡ a −2 + 2a, B ≡ −ra 2 + (r 2 − s)a, and C ≡ a 2 + 2a −1. An integer n is said to have an Isignature if its signature mod n is congruent to (r, s, D ′ , D, r, s), where D ′ + D ≡ rs − 3 mod n and (D ′ − D) 2 ≡ ∆. Definition. A Perrin pseudoprime with parameters (r, s) is an odd composite n such that either
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
ON THE SOLUTIONS TO (n) = (n + k)
"... Abstract. We study the number and nature of solutions of the equation (n) = (n + k), where denotes Euler's phifunction. We exhibit some families of solutions when k is even, and we conjecture an asymptotic formula for the number of solutions in this case. We show that our conjecture follows f ..."
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Abstract. We study the number and nature of solutions of the equation (n) = (n + k), where denotes Euler's phifunction. We exhibit some families of solutions when k is even, and we conjecture an asymptotic formula for the number of solutions in this case. We show that our conjecture follows from a quantitative form of the prime ktuples conjecture. We also show that the prime ktuples conjecture implies that there are arbitrarily long arithmetic progressions of equal values. 1.
Anatomy of Integers and Cryptography
, 2008
"... It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and under ..."
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It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and understanding of the multiplicative structure of “typical ” integers and also “typical ” terms of various sequences such as shifted primes, polynomials, totients and so on. Part I