Introduction to Prime Numbers

A prime number or prime for short is defined as a number that is divisible only by 1 and itself. A number that is not prime is called composite. By definition, 1 is neither prime nor composite. Prime numbers are natural numbers, which are a fancy way of saying positive whole numbers. The first few prime numbers are 2, 3, 5, 7, 11, 13,…

Prime numbers are extremely important not just in mathematics, but in sciences related to mathematics. One of the most important use of prime numbers that affects daily life, but goes unnoticed deals with encryption. When an email is sent it is often encrypted and then decrypted upon receipt. Algorithms are used that employ prime numbers to deceiver the messages. The bigger the prime number used, the harder and longer the decryption process takes.

It has been recently proven that there are an infinite number of primes. Finding a new prime number is a difficult process that cannot be done by hand. A super computer is needed because a potential number is divided by every number smaller than it to see if it is in fact prime. This can take days, months, or even years.

Another way of thinking about prime numbers are that primes cannot be factored, or primes can only be factored trivially, 1 and itself.

Prime numbers play a key role in the Fundamental Theorem Of Arithmetic. This theorem is a corollary of the first of Euclid’s Theorems. The Fundamental Theorem Of Arithmetic basically states: every number can be expressed as the unique product of primes. For instance, the number 147 = 3x7x7. 3 and 7 are both prime numbers. There is no other way to express 147 as the product of primes. This is an incredible discovery and has tremendous advanced implications.

Similar to the normal prime numbers is a concept known as relatively prime. Two numbers are relatively prime if the two numbers share no common divisors except 1. For instance, the following set {3, 4, 9}, has 3 and 4 and 4 and 9 being relatively prime, since they share no common divisors other than 1. Although 4 is not usually considered prime, in this given set, 4 is relatively prime to both 3 and 9. 3 and 9 are not relatively prime because they share a common divisor of 3.

Prime numbers are very important numbers especially in the computer and encryption area. It is a relatively easy concept to understand, but can become complicated quickly. Prime numbers have fascinated the best of the best for centuries and are constantly and area of frequent study.