Integers with a large friable component

Integers with a Large Smooth Divisor

We study the function Θ(x, y, z) that counts the number of positive integers n ≤ x which have a divisor d > z with the property that p ≤ y for every prime p dividing d. We also indicate some cryptographic applications of our results.

Probability That the K-gcd of Products of Positive Integers Is B-friable

In 1849, Dirichlet [5] proved that the probability that two positive integers are relatively prime is 1/ζ(2). Later, it was generalized into the case that positive integers has no nontrivial kth power common divisor. In this paper, we further generalize this result: the probability that the gcd of m products of n positive integers is B-friable is ∏ p>B [ 1− { 1− ( 1− 1 p )n}m] for m ≥ 2. We sho...

Arithmetic of Large Integers

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Methods of Factoring Large Integers

Integers with A

In this note we prove only the important special case (1.1), omitting the parts of the argument required for other cases. In addition, we present an alternate proof, dating from 2002, of the lower bound implicit in (1.1). This proof avoids the use of results about uniform order statistics required in [3], and instead utilizes the cycle lemma from combinatorics. Although shorter and technically ...