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prime-checker

A prime number checker tests whether a given positive integer is prime — divisible only by 1 and itself — and for composites, returns the full prime factorization. The ZTools Prime Checker uses trial division for small numbers and the deterministic Miller-Rabin primality test for large numbers up to 10¹⁵, making it instant for typical inputs and fast even for 15-digit values that would take a naive algorithm minutes.

Use cases​

Cryptography and number theory study​

RSA and other public-key systems rely on the difficulty of factoring large primes. Students experiment with primes of various sizes to understand why a 2048-bit key is hard to crack.

Math homework — prime factorization​

Students enter 360 and see 2³ × 3² × 5. The factorization breakdown teaches the method and verifies hand calculations.

Programming exercise validation​

Algorithm-practice problems often involve primes. Validate "is 982451653 prime?" without writing your own primality test.

Generating large prime test cases​

Need a prime around 10¹²? Increment until the checker says yes. Useful for unit-test fixtures and benchmark inputs.

How it works​

  1. Enter a positive integer — Any value from 2 up to 10¹⁵ (1 quadrillion). Numbers larger than that are still tested but slower; numbers under 2 are not prime by definition.
  2. The checker picks an algorithm — Up to ~10⁶: trial division is fastest. Above that: deterministic Miller-Rabin with carefully chosen witnesses guarantees correctness up to 3.3 × 10¹⁴.
  3. Read the verdict — Either "Prime" or "Composite". For composites, the full prime factorization is returned (e.g., 360 = 2³ × 3² × 5).
  4. See nearby primes — For composites, the next prime above and the previous prime below are also shown — handy for "find a prime near N" tasks.

Examples​

Input: 17

Output: Prime

Input: 360

Output: Composite — 2³ × 3² × 5

Input: 982451653

Output: Prime

Frequently asked questions​

What's the definition of a prime number?

A natural number greater than 1 that has exactly two divisors: 1 and itself. So 2, 3, 5, 7, 11, 13… are prime. 1 is not prime by convention; 0 and negative numbers aren't prime by definition.

How does the Miller-Rabin test work?

It picks random "witnesses" and checks an algebraic identity that all primes satisfy. A composite has a high chance of failing for at least one witness. With carefully chosen deterministic witnesses, the test is provably correct for all integers up to 3.3 × 10¹⁴.

Why is 1 not prime?

The unique factorization theorem states every integer > 1 has a unique prime factorization. If 1 were prime, you could include any number of 1s in the factorization, breaking uniqueness. So mathematicians defined 1 as neither prime nor composite.

How fast can primes be tested?

For numbers below 10¹⁵, milliseconds. For arbitrary 1024-bit numbers (RSA key territory), Miller-Rabin still runs in milliseconds with very high probability of correctness.

Are there infinitely many primes?

Yes — Euclid proved this around 300 BC. The proof is by contradiction: assume a finite list, multiply them, add 1, the result has a prime factor not in the list. The Prime Number Theorem refines this to estimate density.

Tips​

  • For "is this prime?" tests up to 10⁹, trial division up to √n is enough.
  • A number ending in 0, 2, 4, 5, 6, or 8 (other than 2 and 5) is never prime.
  • Sum of digits: divisible by 3 → number divisible by 3 → not prime (except 3 itself).
  • For cryptographic primes (256-bit and up), use a vetted library — never roll your own.

Try it now​

The full prime-checker runs in your browser at https://ztools.zaions.com/prime-checker — no signup, no upload, no data leaves your device.

Open the tool ↗

Last updated: 2026-05-05 · Author: Ahsan Mahmood · Edit this page on GitHub