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Reducing the Space Used by the Sieve of Eratosthenes When Factoring

Samuel Hartman, Jonathan P. Sorenson

Abstract

We present a version of the sieve of Eratosthenes that can factor all integers $\le x$ in $O(x \log\log x)$ arithmetic operations using at most $O(\sqrt{x}/\log\log x)$ bits of space. This is an improved space bound under the condition that the algorithm takes at most $O(x\log\log x)$ time. We also show our algorithm performs well in practice.

Reducing the Space Used by the Sieve of Eratosthenes When Factoring

Abstract

We present a version of the sieve of Eratosthenes that can factor all integers in arithmetic operations using at most bits of space. This is an improved space bound under the condition that the algorithm takes at most time. We also show our algorithm performs well in practice.
Paper Structure (5 sections, 1 theorem, 5 equations, 1 table)

This paper contains 5 sections, 1 theorem, 5 equations, 1 table.

Table of Contents

  1. Introduction
  2. Algorithm Description
  3. Packing Prime Divisors
  4. Compressing Small Primes
  5. Empirical Results

Key Result

Theorem 1

The Sieve of Eratosthenes can be improved to find the complete factorization of all integers $n\le x$ using at most $O(x\log\log x)$ arithmetic operations and $O(\sqrt{x}/\log\log x)$ bits of space.

Theorems & Definitions (1)

  • Theorem 1