Prime Number Calculator
Check if a number is prime with an efficient algorithm.
Enter a positive integer (max 1,000,000)
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Uses: Prime numbers are fundamental in cryptography, number theory, and computer science.
Algorithm: This calculator uses trial division with optimization to efficiently determine if a number is prime.
Why is 1 not considered a prime number?
By modern convention, 1 is not prime because including it would break the Fundamental Theorem of Arithmetic, which states every integer greater than 1 has a unique prime factorization. If 1 were prime, factorizations wouldn't be unique (12 = 2x2x3 = 1x2x2x3 = 1x1x2x2x3...). Historically, 1 was sometimes considered prime, but mathematicians excluded it by the early 20th century.
Why is 2 the only even prime number?
Every even number greater than 2 is divisible by 2, so it has at least three divisors (1, 2, and itself), disqualifying it from being prime. 2 itself is prime because its only divisors are 1 and 2. This makes 2 unique as the smallest and only even prime number.
How are prime numbers used in cryptography?
RSA encryption relies on the fact that multiplying two large primes is easy, but factoring their product is computationally infeasible. A typical RSA key uses primes with 300+ digits. Primes are also used in Diffie-Hellman key exchange and elliptic curve cryptography, forming the foundation of secure internet communications.
Are there infinitely many prime numbers?
Yes, Euclid proved this around 300 BCE with an elegant proof by contradiction: if there were finitely many primes, multiplying them all and adding 1 gives a number not divisible by any known prime, contradicting the assumption. We now know there are infinitely many primes, though they become sparser as numbers grow larger.