Prime Numbers (Definition, Examples & Properties)

This post is also available in: हिन्दी (Hindi)

Prime numbers have been studied for thousands of years. Euclid’s ‘Elements’, published about 300 B.C., proved several results about prime numbers. What is a prime number and what is special about these numbers?

The numbers such as $5$, $7$, or $11$ are called prime numbers as all these numbers have only $2$ factors, viz., $1$ and the number itself.

In this article, let’s learn about prime numbers and their properties.

What Are Prime Numbers?

A number greater than $1$ with exactly two factors, i.e., $1$ and the number itself is called a prime number. For example, $5$ has only $2$ factors, $1$ and $5$ itself. So, it is a prime number. 

On the other hand, there are numbers that have more than two factors, including $1$ and the number itself.  Such numbers are called composite numbers. For example, $6$ is a composite number as it has $4$ factors, $1$, $2$, $3$ and $6$. All such numbers which are not prime numbers are called composite numbers.

Note

• $1$ is the lowest factor of every number.
• The number itself is the largest factor of any number.

Prime Numbers From $1$ to $1000$

Here is the list of the first $1000$ prime numbers

$\lt 10$	$2$, $3$, $5$, $7$
$10 \lt 20$	$11$, $13$, $17$, $19$
$20 \lt 30$	$23$, $29$
$30 \lt 20$	$31$, $37$
$40 \lt 30$	$41$, $43$, $47$
$50 \lt 40$	$53$, $59$
$60 \lt 50$	$61$, $67$
$70 \lt 60$	$71$, $73$, $79$
$80 \lt 70$	$83$, $89$
$90 \lt 100$	$97$
$100 \lt 200$	$101$, $103$, $107$, $109$, $113$, $127$, $131$, $137$, $139$, $149$, $151$, $157$, $163$, $167$, $173$, $ 179$, $181$, $191$, $193$, $197$, $199$
$200 \lt 300$	$211$, $223$, $227$, $229$, $233$, $239$, $241$, $251$, $257$, $263$, $269$, $271$, $277$, $281$, $283$, $293$
$300 \lt 400$	$307$, $311$, $313$, $317$, $331$, $337$, $347$, $349$, $353$, $359$, $367$, $373$, $379$, $383$, $389$, $397$
$400 \lt 500$	$401$, $409$, $419$, $ 421$, $431$, $433$, $439$, $443$, $449$, $457$, $461$, $463$, $467$, $479$, $487$, $ 491$, $499$
$500 \lt 600$	$503$, $509$, $521$, $523$, $541$, $547$, $557$, $563$, $569$, $571$, $577$, $587$, $593$, $599$
$600 \lt 700$	$601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $ 691$
$700 \lt 800$	$701$, $709$, $719$, $ 727$, $733$, $739$, $743$, $751$, $757$, $761$, $769$, $773$, $787$, $797$
$800 \lt 900$	$809$, $811$, $821$, $823$, $827$, $829$, $839$, $853$, $857$, $859$, $863$, $877$, $881$, $883$, $887$
$900 \lt 1000$	$907$, $911$, $919$, $929$, $937$, $941$, $947$, $953$, $967$, $971$, $977$, $983$, $991$, $997$

Methods of Finding Prime Numbers

There are various methods of finding prime numbers. Some of these are discussed as follows.

Sieve Method

Before calculators and computers, numerical tables were used for recording all of the primes or prime factorizations up to a specified limit and are usually printed. The most beloved method for producing a list of prime numbers is called the Sieve of Eratosthenes. This method results in a chart called the Eratosthenes chart, as given below. 

The steps involved in separating the prime numbers from $1$ to $100$ are as follows:

Step 1: First, write all the natural numbers from $1$ to $100$, row-wise and column-wise, as shown in the figure below

Step 2: Put a cross over $1$, as it is neither a prime number nor a composite.

Step 3: Now, encircle the number $2$ (which is a prime number) and cross all the multiples of $2$, such as $4$, $6$, $8$, $10$, $12$, and so on. Since all the multiples of $2$ are composite

Step 4: Next, encircle the number $3$, and put a cross over all the multiples of $3$, such as $9$, $15$, $21$, etc. All the multiples from $3$, except $3$, are composite numbers

Step 5: Again, encircle the number $5$ (since it has only two factors), and put a cross over all the multiples of $5$

Step 6: Now encircle $7$ and cross all the multiples of $7$

Step 7: Continue the process unless all the numbers are either encircled or crossed

The chart below shows the prime numbers up to $100$, represented in coloured boxes.

Using an Algebraic Expression

One can find prime numbers using algebraic expressions. There are many such expressions. Two of these expressions are as follows.

Expression 1

The algebraic expression $n^{2} + n + 41$, where $n$ is any whole number, can be used to find prime numbers greater than $40$. Let’s substitute some values to get the prime numbers.

Substituting $n = 0$ in  $n^{2} + n + 41$, we get $0^{2} + 0 + 41 = 0 + 0 + 41 = 41$

Substituting $n = 1$ in  $n^{2} + n + 41$, we get $1^{2} + 1 + 41 = 1 + 1 + 41 = 43$

Substituting $n = 2$ in  $n^{2} + n + 41$, we get $2^{2} + 2 + 41 = 4 + 2 + 41 = 47$

Substituting $n = 3$ in  $n^{2} + n + 41$, we get $3^{2} + 3 + 41 = 9 + 3 + 41 = 53$

All these numbers $41$, $43$, $47$ and $53$ are prime numbers.

Expression 2

The algebraic expression $6n + 1$ or $6n – 1$, where $n$ is any whole number, can be used to find prime numbers greater than $3$. Let’s substitute some values to get the prime numbers.

Substituting $n = 1$ in  $6n + 1$, we get $6 \times 1 + 1 = 6 + 1 = 7$ is a prime number.

Substituting $n = 1$ in  $6n – 1$, we get $6 \times 1 – 1 = 6 – 1 = 5$ is a prime number.

Substituting $n = 2$ in  $6n + 1$, we get $6 \times 2 + 1 = 12 + 1 = 13$ is a prime number.

Substituting $n = 2$ in  $6n – 1$, we get $6 \times 2 – 1 = 12 – 1 = 11$ is a prime number.

All these numbers $5$, $7$, $11$, and $13$ are prime numbers.

Prime vs Composite Numbers

In a set of natural numbers, the numbers are either prime numbers or composite numbers.

Prime Numbers

A prime number is a number greater than that has exactly $2$ factors, $1$, and the number itself. The smallest prime number is $2$ and there doesn’t exist any largest prime number as there are infinite prime numbers. All the prime numbers are odd numbers except $2$. Examples of prime numbers are $2$, $3$, $5$, $7$, $11$, …

Composite Numbers

A composite number is a number greater than that has more than  $2$ factors, including $1$, and the number itself. The smallest composite number is $4$ and there doesn’t exist any largest composite number as there are infinite composite numbers. A composite number can be an odd number or an even number. Examples of composite numbers are $4$, $6$, $8$, $9$, and $10$, …

Is $1$ a Prime Number?

The number of factors that $1$ has is $1$. Thus, it doesn’t qualify the definition of prime numbers (having $2$ factors) and also doesn’t qualify the definition of composite numbers (having more than $2$ factors). Thus, $1$ is neither a prime number nor a composite number. It’s a unique number.

Twin Prime Numbers

The prime numbers with only one composite number between them are called twin prime numbers or twin primes. The other definition of twin prime numbers is the pair of prime numbers that differ by $2$ only. For example, $3$ and $5$ are twin primes because $5 – 3 = 2$.

The other examples of twin prime numbers are:

• $\left(5, 7 \right) \{7 – 5 = 2 \}$
• $\left(11, 13 \right) \{13 – 11 = 2 \}$
• $\left(17, 19 \right) \{19 – 17 = 2 \}$
• $\left(29, 31 \right) \{31 – 29 = 2 \}$
• $\left(41, 43 \right) \{43 – 41 = 2 \}$
• $\left(59, 61 \right) \{61 – 59 = 2 \}$
• $\left(71, 73 \right) \{73 – 71 = 2 \}$

Co Prime Numbers

Two numbers are called coprime to each other if their highest common factor is $1$. For example, $2$ & $3$ and $6$ and $13$ are co prime because the common factor is $1$ only. Here $2$ and $3$ are twin primes also. But $3$ and $6$ are not twin primes.

Conclusion

The prime numbers are the numbers greater than $1$ that have exactly $2$ factors, $1$ and the number itself. Any other number greater than $1$ that is not a prime number is a composite number. Both the prime numbers and composite numbers are infinite and there exists no greatest prime or composite number.

Practice Problems

State True or False

• A prime number has only  $1$ factor.
• A composite number has only  $1$ factor.
• A prime number has only  $2$ factors.
• A composite number has only  $2$ factors.
• A prime number has more than $2$ factors.
• A composite number has more than $2$ factors.
• $1$ is a prime number.
• $1$ is a composite number.
• $1$ is a unique number.
• $1$ has only $1$ factor.
• $1$ has $2$ factors.
• $1$ has more than $2$ factors.
• $2$ is the smallest prime number.
• $4$ is the smallest prime number.
• $2$ is the smallest composite number.
• $4$ is the smallest composite number.

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties

FAQs