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21
Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... 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 res ..."
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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.
Some problems on the prime factors of consecutive integers
 Illinois J. Math
, 1967
"... 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 eve ..."
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Cited by 9 (0 self)
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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) Zt 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
Approximating the number of integers without large prime factors
 Mathematics of Computation
, 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting Dickman’s function. 1.
Fast Bounds on the Distribution of Smooth Numbers
, 2006
"... Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our fi ..."
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Cited by 3 (2 self)
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Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.
Irregularities in the distribution of primes in function fields, J. Number Theory, accepted for publication
"... Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1. ..."
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Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1.
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
On the Least Prime in Certain Arithmetic Progressions
"... We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. gr ..."
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We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the kfold iterated logarithm, fl is Euler's constant, and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm...
Elliptic Curve Factorization Using a "Partially Oblivious" Function.
"... . Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be nontrivial and that f(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Intr ..."
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. Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be nontrivial and that f(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Introduction It is not known how to eciently factor a large integer N . Currently, the algorithm with best asymptotic complexity is the Number Field Sieve (see [6] ). For numbers below a certain size (currently believed to be about 100 decimal digits), either the Quadratic Sieve [12] or Lenstra's Elliptic Curve Method (ECM) [7] are faster. Which of these algorithms to use depends on the size of N and of the smallest prime factor of N . When the size of the smallest factor is suciently smaller than p N , ECM is the fastest of the three. This note describes a speedup of ECM under special conditions. Suppose N = P R, where P is a prime not dividing R. We assume the size, in bits, of P is know...