# Sieve of Eratosthenes (Prime List Generator)

Enumerate primes up to 10⁷ (or within [m, n]) with the classic sieve; shows base primes, crossing-out operations, exact counts, and up to 200 primes.

> Canonical page: https://elysiatools.com/en/tools/sieve-of-eratosthenes

- **Category:** Math & Numbers

- **Keywords:** sieve of eratosthenes, prime numbers, prime list, prime generator, prime counting, composite numbers, number theory

## Overview

The Sieve of Eratosthenes generator enumerates prime numbers up to 10⁷ or within a custom interval [m, n]. It calculates prime counts, identifies base primes, and tracks composite cross-out operations to demonstrate the inner mechanics of the classic algorithm.

## Inputs

- **Upper limit n** (text): Sieve bound, 2 ≤ n ≤ 10⁷ (byte-sieve limit).
- **Lower bound m (optional)** (text): Only list primes ≥ m; leave empty to enumerate from 2.

## When to use

- You need an exact count of primes (π(n)) or a list of prime numbers up to 10⁷.
- You want to inspect prime numbers within a specific range \[m, n\].
- You are studying or teaching number theory and want to see base primes (p² ≤ n) and composite cross-out counts.

## How it works

- Enter an upper sieve bound n between 2 and 10⁷, and optionally enter a lower bound m.
- The algorithm sifts numbers by finding base primes whose squares do not exceed n (p² ≤ n).
- Multiples of each base prime are marked as composites through iterative cross-out operations.
- The tool outputs the list of surviving primes (up to 200 items), total prime counts, and cross-out operation statistics.

## Use cases

- Verifying prime density and calculating the prime counting function π(n) for math homework or research.
- Generating prime lists within specific numerical ranges for cryptographic and hashing experiments.
- Analyzing the algorithmic efficiency of the Sieve of Eratosthenes by comparing composite counts against total cross-out operations.

## Frequently asked questions

### What is the maximum limit for n in this tool?

The upper bound n can be set up to 10,000,000 (10⁷).

### How many primes are displayed in the output?

The output displays up to 200 prime numbers, but the prime counting function π(n) remains exact regardless of list length.

### What does the optional lower bound m do?

It filters the final prime list to only show primes greater than or equal to m, while the sieve continues to compute from 2 up to n.

### What are base primes in the output?

Base primes are primes satisfying p² ≤ n whose multiples must be crossed out to identify all composite numbers up to n.

### Why does the tool show crossing-out operation counts?

It quantifies the work done by each base prime to show how composites are systematically marked and removed.

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