Sieve of Eratosthenes (Prime List Generator)

Sieve of Eratosthenes (Prime List Generator) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text
Output type: text
Sample coverage: 4
API ready: Yes

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.

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

1Enter an upper sieve bound n between 2 and 10⁷, and optionally enter a lower bound m.
2The algorithm sifts numbers by finding base primes whose squares do not exceed n (p² ≤ n).
3Multiples of each base prime are marked as composites through iterative cross-out operations.
4The 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.

Examples

1. Listing primes up to 100

Math Student

Background

A student learning elementary number theory needs to visualize how the Sieve of Eratosthenes identifies primes under 100.

Problem

Determine all prime numbers ≤ 100 and understand which base primes mark all composite numbers.

How to use

Enter 100 in the 'Upper limit n' field and leave 'Lower bound m' empty.

n: 100

Outcome

Identifies base primes 2, 3, 5, and 7, performs 104 cross-out operations marking 74 composite numbers, and outputs 25 primes ending at 97.

2. Finding primes in a specific range

Algorithm Developer

Background

A developer needs to find the largest prime number under 1000 to construct a hash table bucket size.

Problem

Quickly inspect prime numbers between 997 and 1000 without manually checking factorizations.

How to use

Set 'Upper limit n' to 1000 and 'Lower bound m' to 997.

FAQ

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.

n: 1000, m: 997

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Listing primes up to 100

2. Finding primes in a specific range

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