liminfoPrime Number Tools
Free web tool: Prime Number Tools
About Prime Number Tools
Prime Number Tools is a free browser-based calculator providing five distinct prime number functions: primality testing, prime factorization, listing primes up to N (using the Sieve of Eratosthenes), finding the Nth prime, and counting how many primes exist up to a given value. All calculations run entirely in JavaScript on your device — no server, no delay.
Students, mathematicians, cryptographers, software engineers, and competitive programmers use prime number tools regularly. Primes are the building blocks of the integers and are fundamental to modern cryptography (RSA, Diffie-Hellman), computer science algorithms, and number theory research. This tool covers the most common prime-related operations needed in education and practical computing.
Each function uses an efficient algorithm appropriate to its task. Primality testing uses trial division with 6k±1 optimization for O(√n) performance. Factorization uses the same optimized trial division approach up to 1 billion. The prime list uses a classic bit-array Sieve of Eratosthenes up to 1 million. The Nth prime finder iterates with the isPrime check up to 100,000. The prime counting function applies the sieve up to 10 million for exact results.
Key Features
- Primality test: determine if a number is prime; if not, immediately shows its complete prime factorization
- Prime factorization: decompose any integer up to 1,000,000,000 into prime factors with exponents and expanded form
- List primes up to N: generate all primes up to 1,000,000 using the Sieve of Eratosthenes, showing count and comma-separated list
- Nth prime finder: compute the exact Nth prime number for any N up to 100,000
- Prime counting function (π(n)): count the exact number of primes less than or equal to N, up to 10,000,000
- Copy result button for all modes — copy the full output including factorization or prime list to clipboard
- Input validation with clear error messages for out-of-range values specific to each function's limits
- 100% client-side JavaScript — all calculations run in the browser with no network requests
Frequently Asked Questions
What is a prime number?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. By convention, 1 is not prime. The number 2 is the only even prime; all other even numbers are divisible by 2 and therefore composite.
How does the primality test work?
The tool uses trial division with a 6k±1 optimization. It first checks divisibility by 2 and 3, then tests all numbers of the form 6k-1 and 6k+1 up to √n. This is based on the fact that all primes greater than 3 are of the form 6k±1. The algorithm runs in O(√n) time, making it efficient for numbers up to billions.
What is prime factorization?
Prime factorization (also called integer factorization) decomposes a composite number into a product of prime numbers. For example, 360 = 2³ × 3² × 5. The factorization is unique for every integer greater than 1 (Fundamental Theorem of Arithmetic). This tool shows both the compact exponent form and the expanded multiplication form.
What is the Sieve of Eratosthenes?
The Sieve of Eratosthenes is an ancient algorithm for finding all primes up to a given limit. It works by iteratively marking the multiples of each prime starting from 2 as composite. Numbers that remain unmarked are prime. The tool implements this with a boolean array, running in O(n log log n) time — extremely efficient for finding all primes up to 1,000,000.
What is the prime counting function π(n)?
The prime counting function π(n) (pi of n) gives the number of prime numbers less than or equal to n. For example, π(10) = 4 because there are 4 primes (2, 3, 5, 7) up to 10. The prime number theorem approximates π(n) ≈ n / ln(n), but this tool computes the exact count using the Sieve of Eratosthenes for n up to 10,000,000.
What are the input limits for each function?
Primality test: any non-negative integer (practically unlimited for numbers up to billions). Prime factorization: up to 1,000,000,000 (one billion). List primes up to N: N up to 1,000,000 (one million). Nth prime: N up to 100,000 (the 100,000th prime is 1,299,709). Prime counting function: up to 10,000,000 (ten million).
Why are prime numbers important in cryptography?
Modern public-key cryptography (RSA, DSA, Diffie-Hellman) relies on the computational difficulty of factoring large numbers. The RSA algorithm works because multiplying two large primes is fast, but factoring their product into the original primes is computationally infeasible for large enough numbers (typically 2048-bit or larger). This asymmetry underpins the security of HTTPS, digital signatures, and encrypted communications.
Is there a largest known prime?
There are infinitely many primes (proven by Euclid around 300 BCE). However, the largest known prime is always the result of the latest distributed computing search. As of 2024, the largest known prime is a Mersenne prime (2^136,279,841 - 1) discovered by the GIMPS (Great Internet Mersenne Prime Search) project. This tool works with primes up to 10 million for practical calculations.