A comprehensive prime number tool: primality testing, Nth prime finder, prime range listing, prime factorization, and Sieve of Eratosthenes visualization.
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, 13. 2 is the only even prime number; all other primes are odd.
1 does not satisfy the definition of a prime number, which requires being greater than 1. Additionally, including 1 as a prime would break the uniqueness of prime factorization (Fundamental Theorem of Arithmetic), so mathematicians exclude it.
Yes, there are infinitely many prime numbers. This was proven by Euclid around 300 BC using proof by contradiction. If primes were finite, multiplying all of them and adding 1 would produce a number not divisible by any known prime, leading to a contradiction.
The Sieve of Eratosthenes is a prime-finding algorithm devised by the ancient Greek mathematician Eratosthenes. Starting from 2, it repeatedly marks multiples of each prime, and the remaining numbers are prime. It is the oldest and most intuitive prime-finding algorithm, very efficient for small ranges.
Prime numbers are one of the most fundamental concepts in mathematics, playing a crucial role in modern computer science including cryptography (RSA), hash functions, and random number generation. The difficulty of factoring large composite numbers made from two large primes forms the basis of public-key cryptosystems.
The distribution of primes is described by the Prime Number Theorem. The count of primes below N is approximately N/ln(N), and prime density gradually decreases as numbers grow. However, many unsolved problems about primes remain, including the Twin Prime Conjecture and Goldbach's Conjecture.