Prime numbers

Prime Numbers are all whole Numbers greater than 1 that can only be divided by 1 or by the Number itself.

 

Examples of Prime Numbers

The first Prime Numbers are:

 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …

 

Note that 2 is the only even Prime Number, because all other even Numbers can be divided by 2 and are therefore Composite.

 

 

Composite Numbers

• The Number 12 is not a Prime Number because it can be divided by 1, 2, 3, 4, 6, 12. These kinds of Numbers are called Composite Numbers.
• The Number 13 is on the other hand a Prime Number because it can only be divided by 1 and 13 (itself).

Another example is 21, which is not a Prime Number because it can be divided by 3 and 7. Both 3 and 7 are Prime Numbers and thus Prime Factors of 21.

 

 

Prime Factorization

All Composite Numbers can be divided into a series of Prime Numbers multiplied together. This division is called Prime Factorization.

For example, 40 can be divided into its Prime Factors:

 

\(\large 40 = 2 \cdot 2 \cdot 2 \cdot 5 = 2^3 \cdot 5 \)

 

Another example is 84:

 

\(\large 84 = 2 \cdot 2 \cdot 3 \cdot 7 = 2^2 \cdot 3 \cdot 7 \)

 

One can therefore say that Prime Numbers are the building blocks of all the Natural Numbers.

 

 

The Fundamental Theorem of Arithmetic

Every positive whole Number greater than 1 is either itself a Prime Number or can be written uniquely as a product of Prime Numbers.

This is called the Fundamental Theorem of Arithmetic and is one of the cornerstones of Number Theory.

 

 

How to test for Prime Numbers?

To check if a Number is a Prime Number, one only needs to test division by smaller Prime Numbers up to the square root of the Number.

For example, to check if 29 is a Prime Number, we only need to test division by the Prime Numbers 2, 3 and 5, because \(\large \sqrt{29} \approx 5.38\). None of them divide into 29, so 29 is a Prime Number.

 

 

Special types of Prime Numbers

Mathematicians have given names to different types of Prime Numbers:

• Twin Primes: Two Prime Numbers with the difference of 2, for example 11 and 13.
• Mersenne Primes: Of the form \(\large 2^p - 1\), where \(\large p\) itself is a Prime Number, for example \(\large 2^3 - 1 = 7\).

 

 

Prime Numbers in history and today

Prime Numbers have been studied since antiquity. The Greek mathematician Euclid showed more than 2000 years ago that there are infinitely many Prime Numbers.

Today, Prime Numbers play a central role in modern technology. Large Prime Numbers are for example used to encrypt data and secure communication on the Internet (RSA encryption).