What is Prime Number Checker?

Instantly determine whether any number is prime. For composite numbers, see the complete prime factorization. Also generate lists of primes in any range, find the next/previous prime, and explore prime patterns — useful for math education, cryptography basics, and number theory.

There is no input cap — numbers within the JavaScript safe-integer range take the fast trial-division path, and anything larger falls through to a deterministic BigInt Miller-Rabin test, so cryptographic-scale primes are checked accurately. The factorization tree shows each step (84 → 2 × 42 → 2 × 21 → 3 × 7); for very large composites a leftover cofactor above 10^7 is shown rather than split further. The range finder caps at 10 million entries to keep results scrollable, and you can jump straight to the Nth prime for programming challenges and number theory exercises.

How to use

1. Enter any positive integer to check if it's prime — results are instant even for large numbers.
2. For composite numbers, view the complete prime factorization tree. For primes, see the nearest primes above and below.
3. Use the range finder to list all primes between two numbers, or generate the first N primes.

When to use

• Verifying a homework answer in number theory or discrete mathematics class.
• Picking a prime modulus for a hash function or a toy cryptographic example.
• Generating the first N primes for a Project Euler or competitive programming task.

Result

Enter 97 and it's prime. Enter 84 and it's not, with factors 2² × 3 × 7. Generate all primes between 1 and 100 to see all 25 in that range.

FAQ

How fast can the tool check a very large number?

Well under 10 milliseconds for typical inputs. Miller-Rabin with a deterministic witness set runs in roughly O(log³ n) modular exponentiations, so even a number with dozens of digits finishes in a few dozen multiplications. The UI feels instant whether you check a 12-digit number or a cryptographic-scale prime.

Is the prime factorization always unique?

Yes. The fundamental theorem of arithmetic guarantees that every integer greater than 1 has exactly one prime factorization, up to ordering. The tool sorts factors ascending and groups duplicates with exponents (84 = 2² × 3 × 7) to match textbook notation.

What is the largest number the tool will check?

There is no fixed limit. Numbers within the JavaScript safe-integer range use the faster trial-division path, and larger inputs are handled by a BigInt Miller-Rabin test. That test is provably exact for every number below 3.3 × 10^24 and stays an extremely strong probable-prime test well beyond that, so it comfortably checks cryptographic-scale primes.

Why are 0 and 1 not counted as primes?

A prime is defined as a positive integer with exactly two distinct divisors, 1 and itself. The number 1 has only one divisor (itself), so it doesn't qualify; 0 has infinitely many divisors. Excluding both makes unique factorization hold without special cases.

How do primes relate to real-world cryptography?

RSA picks two primes around 1024 bits each, multiplies them, and publishes the product as part of its public key. The hardness of factoring that product back into the two primes is what keeps RSA secure. Primality testing itself remains fast, even at those sizes.

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