What are Prime Numbers?

Prime Numbers Definition

All the natural numbers greater than 1 that have only two factors one and the number itself are called Prime numbers.

In other words, a Natural Number n greater than 1 is prime if it has only two factors 1 and n.

Prime Numbers Examples

Some initial prime numbers in ascending order are

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, . . .

All the above numbers have only two factors. They are only divisible by 1 and itself.

Note:

• 1 is not a Prime Number because it has only factor 1 which is itself only.
• 2 is the only even Prime Number.

Types of Prime Numbers

• Even Prime Numbers: The numbers that are even and prime. 2 is the only even prime number.
• Odd Prime Numbers: The numbers that are odd and prime are Odd Prime Numbers. There are infinitely many such numbers, in fact, all prime numbers except 2 are odd prime numbers. Example: 3, 5, 7, 11, etc.

Note: A number that is divisible by 2 is known as an Even Number.

Note: A number that is not divisible by 2 is known as an Odd Number.

List of Prime Numbers

Prime Numbers from 1 to 100 are listed below:

List of Prime Numbers from 1 to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 & 97.

Facts about Prime Numbers

1. Infinite: There are infinitely many prime numbers. This fact was first proven by the ancient Greek mathematician Euclid around 300 BCE in his seminal work, “Elements.”
2. Distribution: Prime numbers become less frequent as numbers increase, but they are still infinite in number and are distributed irregularly throughout the number line. This irregularity is captured by the Prime Number Theorem, which provides an asymptotic approximation of the distribution of prime numbers.
3. Twin Primes: Twin primes are pairs of prime numbers that have a difference of 2 between them (e.g., 3 and 5, 11 and 13, 17 and 19). The existence of infinitely many twin primes is a famous unsolved problem in number theory known as the Twin Prime Conjecture.
4. Role in Cryptography: Prime numbers play a central role in modern cryptography, particularly in algorithms like RSA encryption, where the security of the encryption relies on the difficulty of factoring large composite numbers into their prime factors.
5. Mersenne Primes: Mersenne primes are prime numbers that can be expressed in the form 2^p−1, where p is also a prime number. They have applications in number theory and have been extensively studied due to their intriguing properties.
6. Goldbach Conjecture: The Goldbach Conjecture, proposed by the German mathematician Christian Goldbach in 1742, states that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite extensive computational verification for vast ranges of numbers, the conjecture remains unproven.
7. Historical Significance: The study of prime numbers has a rich historical significance, dating back to ancient civilizations such as Greece and Egypt. Prime numbers have been studied for their mathematical properties and have practical applications in various fields, including cryptography, number theory, and computer science.

Prime and Composite Numbers are commonly used classifications of Natural Numbers based on divisibility and the number of Factors. A Prime Number has only two factors while Composite Numbers have more than two factors. This classification of Numbers makes the study of natural numbers more organized and convenient and is useful in a variety of situations like computer algorithms, biology, understanding of Number Theory, etc.

This article describes what are prime and composite numbers, the types of primes and composite numbers, and tests to check whether a given number is prime or not (primality tests). Finally, a few solved questions, and a few practice problems related to prime and composite numbers are presented.