Test whether any integer is prime, identify composite divisors, find nearest prime bounds, and generate prime lists with the Sieve of Eratosthenes.
Click "Check Primality" to test number.
A cybersecurity engineer generates 2048-bit RSA encryption keys by finding two massive prime numbers p and q and computing N = p × q. A computer scientist designs a hash table using a prime bucket array size to minimize hash collisions. A number theorist searches for Mersenne primes of the form 2ᵖ - 1 using Lucas-Lehmer primality tests.
A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. The sequence of prime numbers begins: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37... Number 2 is the only even prime number; all other primes are odd. Integers greater than 1 that are not prime are called composite numbers.
Prime numbers serve as the fundamental building blocks of integers, enshrined in the Fundamental Theorem of Arithmetic: every positive integer greater than 1 can be uniquely factored into a product of prime powers. This tool tests whether any given integer is prime, identifies composite factors, finds nearest prime bounds, and generates prime lists via the Sieve of Eratosthenes. The following sections cover primality algorithms, efficiency bounds up to √n, and modern cryptographic applications.
When an integer n is evaluated, the engine executes trial division up to ⌊√n⌋. For list generation, it constructs an array bitset using the Sieve of Eratosthenes.
1. Small Input Base Cases:
- If n ≤ 1: Not prime (0 and 1 are neither prime nor composite).
- If n = 2 or n = 3: Prime.
- If n % 2 = 0 or n % 3 = 0: Composite (divisible by 2 or 3).
2. Trial Division Limit (6k ± 1 Optimization):
All prime numbers greater than 3 can be expressed as 6k ± 1. The algorithm tests potential divisors i starting at 5 and incrementing by 6 (checking i and i + 2) up to i × i ≤ n.
Limit = ⌊√n⌋. If no divisor is found up to √n, n is guaranteed prime.
3. Sieve of Eratosthenes Algorithm:
To generate all primes up to N:
Create boolean array is_prime[0..N] initialized to true. Set 0 and 1 to false.
For p = 2, 3, 5... up to √N:
If is_prime[p] is true, mark all multiples p², p² + p, p² + 2p... as false.
Public-key cryptography (RSA and ECC). Modern HTTPS security, SSL certificates, and digital signatures rely on the computational difficulty of factoring the product of two large prime numbers.
Hash tables and database indexing. Hash table bucket array capacities are chosen as prime numbers to distribute key hashes uniformly and reduce collision clustering.
Pseudorandom number generation. Linear congruential generators (LCGs) and Diffie-Hellman key exchanges select large prime moduli m to achieve maximum period length before sequence repetition.
Cicada evolutionary survival strategies. Periodical cicadas emerge every 13 or 17 years—both prime numbers—to minimize synchronization with predator population cycles.
For single numbers up to 10¹⁵, Mode 1 delivers instant trial division results in under 1 millisecond.
Use Mode 2 (Sieve Generator) to export complete lists of prime numbers up to 100,000 for coursework, programming competitions, or prime gap analysis.
For prime factor decomposition ($N = p_1^{e_1} \times p_2^{e_2}...$), use our Prime Factorization Calculator. For greatest common divisors, use our GCD Calculator.
The primality engine executes locally in JavaScript using double-precision floating-point trial division and boolean Uint8Array memory allocation for sieves.
| Feature | This Tool | Hand Trial Division | Sieve of Eratosthenes |
|---|---|---|---|
| Trial Division Limit | i ≤ √n (6k ± 1) | Manual division | Array elimination |
| Time Complexity | O(√n) single test | Hours for large n | O(N log log N) list |
| Nearest Primes | Automatic lower/upper bounds | Manual search | Range lookup |
| Max Sieve Range | N = 100,000 (9,592 primes) | N = 50 practical | Memory bound |
| Privacy | Client-side browser | Paper | Local device |
| Cost | Free | Free | Free |
Every even integer greater than 2 is divisible by 2 (e.g. 4 = 2 × 2, 6 = 2 × 3), which violates the definition of a prime number having no divisors other than 1 and itself.
The largest known prime numbers are Mersenne primes of the form 2ᵖ - 1, containing tens of millions of digits discovered by the Great Internet Mersenne Prime Search (GIMPS).
A prime gap is the difference between two consecutive prime numbers (e.g., the gap between 7 and 11 is 4). Twin primes are primes with a gap of 2 (e.g., 11 and 13).
Prime Factorization — Decomposes any integer into prime factor trees and exponent notation.
GCD Calculator — Computes greatest common divisors using the Euclidean algorithm.
LCM Calculator — Computes least common multiples for 2+ numbers.