Prime Number Testing & Factorization Guide
A prime number calculator checks whether a number is prime and performs prime factorization. It uses trial division for small inputs and faster probabilistic tests for large integers.
What is Prime Number Calculator?
The prime number calculator helps students and engineers verify primality, factor small to medium integers, and explore patterns (e.g., twin primes, gaps) with instant feedback.
How to Use the Prime Number Calculator
- Enter an integer (positive).
- Check primality to see if it has any divisors other than 1 and itself.
- Factorize to list prime factors in exponential form.
- Explore neighbors (n-1, n+1) for twins and gaps.
- Copy results for proofs or programming tasks.
Formulas & Methods
- Trial division: test divisors up to
floor(sqrt(n)); skip evens after 2 and use wheel optimizations. - Miller-Rabin (probabilistic): repeat bases to reduce error; for 64-bit
n, a small fixed base set gives deterministic results. - Factorization: combine trial division with pollard rho for mid-size
n. - Fundamental Theorem: every integer
n > 1has a unique prime factorization.
Assumptions & limitations
- Very large
nmay use probabilistic tests; a composite slip is extremely unlikely if multiple bases are used. - Factorization time grows quickly with size; only small to medium
nfactor instantly.
Examples
Example A — Primality
n = 97 → prime (no divisors up to sqrt(97) ~ 9.85).
Example B — Factorization
n = 504 → 2^3 * 3^2 * 7.
| n | Prime? | Factors | |---|---|---| | 37 | Yes | 37 | | 84 | No | 2^2 * 3 * 7 | | 221 | No | 13 * 17 |
Pro Tips & Best Practices
- For proof tasks, show divisor checks up to sqrt(n).
- When factoring, divide out small primes first to shrink n.
- For programming, use Miller-Rabin with a safe base set for 64-bit integers.
- Use modular arithmetic to avoid overflow in big-n tests.
Related Calculators
FAQ
Q: How do you test if a number is prime?
A: Trial division up to sqrt(n) works for small n; faster probabilistic tests like Miller-Rabin are used for large n.
Q: What is prime factorization?
A: Expressing n as a product of prime numbers; unique up to order (Fundamental Theorem of Arithmetic).
Q: Why are primes important?
A: They underpin number theory and practical cryptography (RSA, Diffie-Hellman).
Q: Can the calculator handle big integers?
A: Yes—big integers are supported; primality may use probabilistic tests for speed.
Q: What are twin primes?
A: Pairs of primes that differ by 2, like (11, 13).
Call to Action
Enter a number to test primality or see its factorization—use the tool to learn patterns or verify homework quickly.