Check if a number is prime and generate prime numbers up to any limit with factorization display.
Enter a whole number and the tool says whether it is prime, returning a verdict and, for composite inputs, the prime factorization with exponents. It also generates the complete ascending list of primes at or below any limit you choose. The results that surprise people most are the status of 1, which is neither prime nor composite, and of 2, the only even prime. For composites, read the factorization carefully: a repeated factor appears once with an exponent, so 12 is shown as 2^2 x 3, not as 2 x 2 x 3.
A concrete input and expected output from the current implementation.
Input
360
Expected output
360 is not prime. Prime factorization: 2^3 x 3^2 x 5
360 divides by 2 three times (360 to 180 to 90 to 45), then by 3 twice (45 to 15 to 5), and 5 is prime. Every integer greater than 1 has exactly one prime factorization, so this is the only way to write 360 as a product of primes.
A composite number n always has a factor no larger than the square root of n, because in any product a x b = n at least one factor is at most the square root. Testing divisors only up to that bound is enough: the first exact division reveals a factor and drives the factorization display, and a sweep with no hits means prime. Work scales with the square root of the input, not the input itself.
Generating primes up to a limit works by elimination: list the integers from 2 upward, take the smallest unmarked number as a prime, mark every multiple of it as composite, and repeat until the limit is reached. Whatever remains unmarked is prime, so the output is the complete ascending list of primes at or below the limit. The limit itself appears in the list only when it is prime.
Primality has nothing to do with parity. A prime is an integer greater than 1 with exactly two positive divisors, and 2 has exactly those: 1 and 2. Every other even number is divisible by 2, so it has at least three divisors. That makes 2 the only even prime, and it is also the smallest prime of all.
No. Euclid proved in the Elements (Book IX, Proposition 20) that the primes are infinite, so no generator can print them all; any limit you enter is just a slice of the list. The largest known prime is always a Mersenne prime, of the form 2^p - 1, and the record is retired roughly every couple of years by the GIMPS project, so the current holder is best looked up rather than memorized.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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