Prime numbers and factorisation

A prime number has exactly two divisors: one and itself. Every other number can be written uniquely as a product of primes.

The tool tests primality and shows the full factorisation.

Divisors are tested from 2 up to the square root of the number, dividing whenever the division is exact. Whatever remains at the end, if greater than one, is prime as well.

The divisor count follows from the exponents in the factorisation.

One is not prime, and there is a reason

A prime number has exactly two distinct divisors: one and itself. The number one has only one, so it falls outside by definition. It is not arbitrary: if one were prime, prime factorisation would stop being unique, because you could add as many ones as you liked. All of arithmetic rests on that uniqueness.

They are infinite, and Euclid proved it

The proof is over two thousand years old and remains elegant: if there were only finitely many primes, multiplying them all together and adding one would produce a number divisible by none of them, contradicting the premise.

From curiosity to infrastructure

For centuries they were an intellectual game with no application. Today the security of communications depends on them: encrypting is easy by multiplying two enormous primes, and decrypting without the key would require factorising the product, something no classical computer can do in reasonable time.

They thin out but never run out

The further you go, the further apart they are, but they never stop appearing. The largest known has tens of millions of digits and is found through distributed computing projects.