Definition and Basic Properties
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Key properties include: it is divisible only by 1 and the number itself, making it the building block of all integers through multiplication. The number 2 is the only even prime number, and all other primes are odd.
Unique Factorization and Infinitude
Prime numbers satisfy the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely factored into a product of primes, disregarding order. Additionally, there are infinitely many prime numbers, as proven by Euclid's theorem, ensuring their endless distribution among natural numbers.
Practical Examples
Consider the prime number 7: it has divisors only 1 and 7. For factorization, 30 = 2 × 3 × 5, a unique combination of primes. Testing primality for 13 involves checking divisibility by primes less than its square root (about 3.6), so only 2 and 3; since it is not divisible by them, 13 is prime.
Importance and Applications
Prime numbers form the foundation of number theory and are crucial in cryptography, such as RSA encryption, which relies on the difficulty of factoring large semiprimes. They also appear in sieving methods like the Sieve of Eratosthenes for finding primes and in studying patterns like twin primes.