Algorithms to Uniformly Generate Random Factored Smooth Integers
Eric Bach, Jonathan Sorenson

TL;DR
This paper introduces algorithms for exactly counting, enumerating, and uniformly sampling y-smooth integers up to x, with improvements in efficiency through heuristic methods, and demonstrates their practical application.
Contribution
The paper presents new algorithms for uniform random generation of y-smooth integers with improved runtime and practical implementation details.
Findings
Exact computation of (x,y) and enumeration of y-smooth integers.
A uniform sampling algorithm with O((x,y) loglog y) complexity.
Heuristic-based methods achieving faster runtime of O(( log x)^3 / log extlog x).
Abstract
Let be integers. A positive integer is -smooth if all its prime divisors are at most . Let count the number of -smooth integers up to . We present several algorithms that will generate an integer at random, with known prime factorization, such that is -smooth. We begin by describing algorithms to compute exactly and to enumerate -smooth integers up to in lexicographic order by prime divisor. Both of these are based on Buchstab's identity, and were likely known before. Then we present an algorithm that accepts as input a parameter , , and returns the integer that is at position in the lexicographic ordering of all -smooth integers up to . Here position 0 is the first position. Thus, is generated uniformly so long as is chosen uniformly. This algorithm has a…
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Taxonomy
TopicsAnalytic Number Theory Research · Limits and Structures in Graph Theory · Computability, Logic, AI Algorithms
