Prime Numbers

Definition of Prime Numbers

  • A prime number is a natural number that has exactly two distinct natural number divisors: one and itself.
  • The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, etc.
  • Note that 1 is not a prime number, as it only has one distinct divisor: itself.
  • 2 is the only even prime number. All other prime numbers are odd.

Recognising Prime Numbers

  • Divisibility tests can be used to check if a number is prime. Basically, you divide the number by all numbers (excluding 1 and the number itself) up to its square root. If it’s only divisible by 1 and itself, it’s a prime number.
  • For example, 19 is a prime number because it has no divisors other than 1 and 19.
  • Numbers that have more than two distinct divisors are called composite numbers.

Prime Factorisation

  • Any integer, or whole number, can be broken down into prime factors, a process known as prime factorisation.
  • For example, the prime factorisation of 18 is 2 * 3 * 3 or 2 * 3^2. This means 18 is a composite number, as it has more than two distinct divisors.
  • Prime factorisation is important as it lays the foundation for understanding various concepts in algebra and number theory.

Importance and Application of Prime Numbers

  • Prime numbers play a key role in cryptography, specifically when creating public and private keys.
  • In mathematics, they’re used to construct various proofs and explain certain phenomena.
  • For instance, the Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime number itself or can be factorised as a product of prime numbers, and this factorisation is unique, up to the order of the factors.

Common Misconceptions about Prime Numbers

  • Confusing prime with prime factors: A prime number is a number that has only two distinct divisors, while prime factors are the primes that divide a particular number.
  • Assuming 1 is a prime number: Remember, 1 is not a prime number due to the definition of primes having exactly two distinct divisors.
  • Forgetting 2 is a prime number: Because it is even, people often forget that 2 is a prime number. It’s the only even prime number. All others are odd.