Prime numbers: Definition, Properties, and Methods

Prime numbers are the natural numbers that are only divisible by 1 and the number itself. Prime numbers, in other terms, are positive integers with only two factors: 1 and the number itself. Some of the examples of prime numbers are 2, 3, 5, 7, 11, 13, etc. 1 is neither a composite number nor prime. As a result, every prime number is greater than one. We can also state that, except for 1, the remaining numbers are divided into prime and composite numbers. In this article, we will discuss prime numbers and their properties.

Read More About Natural Numbers and Whole Numbers

Keyterms: Numbers, Natural numbers, Whole numbers, integers, Composite numbers, Factors

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What are Prime Numbers?

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A prime number refers to a positive integer that has two of the same factors. If p is a prime, then the only factors it can have are 1 and p itself. A composite number does not follow this pattern and can be factored into other positive integers. A positive number or integer that is not a product of any other two positive integers other than one and the number itself is another method of defining it. The first 10 prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Note: Number 1 is neither prime nor composite.

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History of Prime Numbers

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The Greek mathematicians such as Euclid (300 BCE) and Eratosthenes of Cyrene (276–194 BCE) examined prime numbers for the first time. Euclid's Elements contained the first known evidence that there exist an unlimited number of primes. Various formulas for discovering primes have been proposed, but they have all been found to be incorrect. The prime number theorem and the Riemann zeta function are two other well-known results relating to prime number distribution.

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How to find prime numbers?

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The two approaches below will assist you in determining whether a particular number is a prime or not.

1. Tests of Divisibility
2. Factorization

Method 1: Tests of Divisibility

To find out if the number is prime or not, some prime number formulas can be used. Follow the steps below to determine whether a number is a prime number or not:

Step 1: Look at the number's unit place. It is not a prime number if it ends in 0, 2, 4, 6, or 8. Numbers ending in 0, 2, 4, 6, and 8 are never prime numbers.

Step 2: Add the digits of that number together. The sum is not a prime number if it is divisible by three.

Step 3: Determine the square root of the given number after checking the result of stages 1 and 2.

Step 4: Simply divide the provided number from all prime numbers below its square root value.

Note: If a large number is ending with 5, then it is always divisible by 5. Hence, it is not a prime number

Method 2: Factorization

Factorization is the most popular approach for finding prime numbers. The following are the stages involved in using the factorization method to find prime numbers:

Step 1: We'll look for the factors of the given integer (factors are the number that completely divides the given number)

Step 2: Next, determine the total number of factors associated with that number.

Step 3: As a result, if the total number of factors is greater than two, the number is a composite number rather than a prime number.

Note: 2 is the only even prime number.

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Prime Numbers vs Composite Numbers

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• Composite numbers are positive integers or whole numbers with more than two divisors, whereas prime numbers have just two components. For example, 19 is a prime number because it has just two factors: 1 and 19 (1 19). 
• The number six, on the other hand, has three divisors: 3,2, and 6 (1 6 and 2 3).
• The divisibility test of orders 2, 5, 3, 11, 7, and 13 is used to determine if a number is prime or composite. A composite number can be divided by any of the factors listed above. A prime number is not divisible by 2, 3, 5, or 7 and is less than 121. The number is composite otherwise. 

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Important Formulas and Examples

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1. The formula below can be used to find prime numbers bigger than 40.

n² + n + 41 (n = 0, 1, 2,....,39)

For example:

(0)² + 0 + 0 = 41

(1)² + 1 + 41 = 43

(2)² + 2 + 41 = 47

2. The only even prime number is 2. The prime numbers 2 and 3 are the only two natural numbers that are prime in consecutive order. Apart from them, every prime number (excluding prime multiples, such as 2, 3, 5, 7, and 11) can be expressed as 6n + 1 or 6n – 1, where n is a natural number.

For example:

6(1) – 1 = 5

6(1) + 1 = 7

6(2) – 1 = 11

6(2) + 1 = 13

6(3) – 1 = 17

6(3) + 1 = 19

6(4) – 1 = 23

6(4) + 1 = 25 (multiple of 5)

Table of Prime Numbers 1 to 200

Prime numbers from 1 to 200 are as tabulated below:

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Properties of Prime Numbers

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• At least one prime number can be divided by any number bigger than one.
• Any even positive integer greater than 2 can be used to represent the sum of two primes.
• All prime numbers except number 2 are odd. 
• Two prime numbers are coprime to each other.
• Each composite number can be factored into prime factors, each of which is unique in its own right.

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Things to Remember

• Prime numbers are the natural numbers that are only divisible by 1 and the number itself.
• The first 10 prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
• The two approaches to determine whether a particular number is a prime or not are Tests of Divisibility and Factorization.
• Composite numbers are positive integers or whole numbers with more than two divisors, whereas prime numbers have just two components.
• 2 is the only even prime number.
• The prime numbers 2 and 3 are the only two natural numbers that are prime in consecutive order.

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