1 is not prime
We keep 1 out of the prime club so every integer >1 has a unique prime factorization (the Fundamental Theorem of Arithmetic).
Definition choice
Prime Number Checker & Prime Factorization
Input & Actions
Tip: accepts integers like 84, -210, or big integers (within reason). Decimals are not supported.
Results will appear here.
Method & Definitions
This tool determines whether a number is prime and computes its prime factorization. It uses optimized trial division up to \( \sqrt{n} \), checking 2, 3, and then only candidates of the form \( 6k \pm 1 \).
- Prime: \(n>1\) with no divisors other than 1 and \(n\).
- Composite: \(n>1\) that is not prime.
- Prime factorization: \( n = \prod p_i^{e_i} \).
\( \text{Prime test: } \nexists d \in \{2,\dots,\lfloor\sqrt{n}\rfloor\} \text{ with } n \bmod d = 0 \)
\( \text{Factorization: } n = \prod p_i^{e_i}, \; e_i \in \mathbb{Z}_{\ge 0} \)
\( \text{Factorization: } n = \prod p_i^{e_i}, \; e_i \in \mathbb{Z}_{\ge 0} \)
FAQ
Is 1 prime?
No. By definition, primes are integers \(\ge 2\).
How are negatives handled?
We display \( -1 \times \) the factorization of \(|n|\). Example: \(-84 = -1 \times 2^2 \times 3 \times 7\).
Performance limits?
Trial division up to \( \sqrt{n} \) is quick for typical classroom sizes (up to ~1010 is usually fine in modern browsers). Extremely large inputs will take longer.
5 Fun Facts about Primes
2 is the only even prime
Every other even number has 2 as a factor, so 2 stands alone—making it both the smallest prime and the loneliest even one.
Even unicorn
Gaps can be huge
Primes get rarer but never stop. There are arbitrarily long stretches of composites—yet another prime always appears eventually.
Prime deserts
6k ± 1 shortcut
Every prime greater than 3 is of the form \(6k \pm 1\). That’s why this tool skips other residues when it searches.
Search trick
RSA leans on factoring
Modern encryption relies on the difficulty of factoring huge semiprimes. Fast factoring breaks keys—hence the race for quantum-safe crypto.
Crypto link