Prime Number Calculator

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... The number 2 is the only even prime. Every non-prime number greater than 1 can be uniquely expressed as a product of prime factors (the Fundamental Theorem of Arithmetic).

The calculator uses trial division up to the square root of the number (since any factor larger than sqrt(n) would pair with a factor smaller than sqrt(n)). For large numbers, it uses optimized wheel factorization. If no divisor is found, the number is prime. The tool handles numbers up to several trillion efficiently.

Prime factorization breaks a number into its prime components. For example: 60 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5. The calculator shows both the expanded form and the exponential form. Every composite number has a unique prime factorization, which is fundamental to number theory and cryptography.

Yes — enter a range (e.g., 1 to 1000) and the calculator lists all prime numbers within it using an optimized Sieve of Eratosthenes algorithm. It also shows the count of primes in the range and the prime density (what percentage of numbers in the range are prime).

Primes are the building blocks of all integers. They are critical in cryptography (RSA encryption relies on the difficulty of factoring large semiprimes), hash functions, random number generation, and error-correcting codes. Understanding primes is also essential in number theory, algebra, and computer science.