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Prime Numbers

Prime Numbers

Prime numbers are one of the coolest things in maths. They have intrigued the mathematicians for thousands of years. The most brilliant of minds have struggled with them in order to fully understand the characteristics of the prime numbers .Despite more than 2000 years of research, we have not been able to comprehend prime numbers fully. So what are prime numbers? Let’s explore.

What is a prime number?
A prime number is a positive integer which has exactly two factors: 1 and itself. For example, let’s take the number 8.We can express the number 8 in the following ways:

8 = 4 x 2
8= 2 x 2 x 2
8= 8 x 1

As we can see, 8 can be expressed in different ways – it is the result that we get when we multiply 4 and 2 or when we multiple three 2s. The numbers 4 and 2, in this case, are known as the factors of 8. It is important to know that as 8 can also be written as the result that we get when we multiply 8 with 1, so it follows that the number 8 itself as well as the number 1 are factors of 8. Now if we take number 7 for example, we can express it only in one way:

7 = 7 x 1

Since no other numbers are factors of 7 apart from 7 itself and 1, 7 is a prime number. And so is 2, 3 , 5, 7 , 11 … the list just goes on.

Is there a biggest prime number?

Many who have just learned about the concept of the prime numbers tend to ask the question : “What is the biggest prime number?”. Well, the answer is there is nothing as such as the biggest prime number. In other words, there is no upper limit on the size of a prime number. If I gave you a very large prime number, you can always prove that there is a bigger prime number. Euclid, a brilliant mathematician of his time and one of the greatest mathematicians of all time, proved this over 2000 years ago.

Is 1 a prime number?
There was a time when 1 was considered as a prime number. However, in the modern times, 1 has lost its “prime” status and is not considered to be a prime anymore. Why? To understand the reason, we must first understand the fundamental theorem of arithmetic. The fundamental theorem of arithmetic, also known as the unique factorization theorem, states that any integer greater than 1 is either a prime or can be expressed as a unique product of prime numbers. Let’s take an example.

Let’s take the number 15. 15 can be expressed as the product of prime numbers in the following way:

15 = 3 x 5

According to the fundamental theorem, this combination of products is unique and 15 cannot be written as the product of any other prime numbers.
Let’s take another example. This time, we take the number 120.

120= 5 x 2 x 2 x 2 x 3

As we can see for 120 also, it can only be written as a product of prime numbers in only one way i.e. using one 5, three 2s and one 3. This is true for any number greater than 1.
If we consider 1 as a prime, then a major problem arises. What is the problem? Let us assume that 1 is a prime. Now, 120 can be written as:
120 = 1 x 5 x 2 x 2 x 2 x 3 or
120 = 1 x 1 x 5 x 2 x 2 x 2 x 3 or
120 = 1 x 1 x 1 x 5 x 2 x 2 x 2 x 3 etc.

As you can see, suddenly, for any number, we can write it as product of prime numbers in many ways i.e. for 120, we could use one 1, one 5, three 2s and one 3. We could also use three 1s, one 5, three 2s and one 3 etc. Furthermore, all these 1s are not adding any value to the information that we already have about a number. As a result, 1 is no longer considered to be a prime any more. Instead, it is considered to be a special case.