# Prime Numbers

## Definition of Prime Numbers

• A prime number is a positive integer greater than 1 that has exactly two distinct divisors: 1 and itself.
• For instance, the numbers 2, 3, 5, 7, and 11 are prime numbers as they only have the divisors 1 and themselves.

## Fundamental Theorem of Arithmetic

• The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 is either a prime number or can be represented as the product of prime numbers, this representation being unique apart from the order of the factors.
• This theorem underscores the importance of prime numbers in number theory and mathematics as a whole.

## Identifying Prime Numbers

• Identifying prime numbers becomes increasingly difficult as numbers get larger. There are several methods for identifying prime numbers, some of the common methods being trial division, the Sieve of Eratosthenes, and various probabilistic tests.
• The Sieve of Eratosthenes is a simple method for finding all prime numbers up to a given limit by crossing off multiples of found prime numbers.

## Properties of Prime Numbers

• Prime numbers all odd except for the number 2, which is the smallest prime number.
• There is no formula that perfectly predicts prime numbers. Their distribution among the positive integers, though well understood in a general sense, is still a matter of much ongoing research.

## Prime Numbers in Cryptography

• Prime numbers play a central role in modern cryptography systems, particularly in public key cryptography such as RSA, where they are used to create ‘trapdoor’ mathematical functions that are easy to compute in one direction but difficult to reverse without knowledge of the correct factors.
• A firm understanding of prime numbers, therefore, forms an important part of studying the mathematical foundations of cryptography.

## Diophantine Equations

• Diophantine equations, named after the ancient Greek mathematician Diophantus, are polynomial equations that seek integer solutions. These equations often involve prime numbers and their properties.
• For example, the equation a^n + b^n = c^n is known as Fermat’s Last Theorem, which conjectures that there are no integer solutions when n is an integer greater than 2.

## Practice Problems

• Practice problem-solving skills with exercises related to prime numbers, including identifying prime numbers, using the Fundamental Theorem of Arithmetic, solving Diophantine equations, and understanding the application of prime numbers in cryptography.
• Deepen understanding of the properties of primes and their patterns by working out problems and puzzles centred around prime numbers. This assists in improving the ability to recognize and work with prime numbers in various contexts.