Definition of Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it cannot be formed by multiplying two smaller natural numbers. For instance, 2 is the smallest prime number because its only divisors are 1 and 2.
Key Properties of Prime Numbers
Prime numbers are fundamental building blocks in number theory. Every integer greater than 1 can be expressed as a product of primes, known as the Fundamental Theorem of Arithmetic. Primes are infinite in quantity, as proven by Euclid, and include both odd numbers like 3 and 5, with 2 being the only even prime.
Examples of Prime Numbers
Common prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, and 23. For example, 7 is prime because it is not divisible by any number from 2 to 6. In contrast, 4 is not prime as it equals 2 × 2, and 9 is composite since 3 × 3 = 9, illustrating the distinction from composite numbers.
Importance and Applications of Prime Numbers
Prime numbers are crucial in fields like cryptography, where algorithms such as RSA rely on the difficulty of factoring large primes for secure data encryption. They also appear in computer science for hashing functions and in pure mathematics for studying divisibility and patterns like the distribution of primes.