One of the most fundamental concepts in mathematics which you must grasp, is that of a prime number. Mathematics and number theory in particular is devoted to their study, as one of the most exciting of research topics.
What are They?
A prime number is any natural number that is fully divisible only by itself and the number 1. In other words, it's a number which cannot be factorized into other numbers. For example, 2 is only divisible by 2 itself and 1. 2 can only be factorized into 2 and 1 (i.e. 2 = 2 x 1). So two is in fact the smallest one of them.
You may ask, why 1 isn't a prime number? Though it fits the definition, by convention it is not considered to be one. It was proved by Euclid, centuries ago, that these numbers are infinite. A composite number is fully divisible by natural numbers, other than itself and 1.
To determine whether a number is a prime or composite, you need to factorize it. If it turns out that it has factors other than itself and 1, it is the latter. If not, then it belongs to the former category.
For example, let us see if 9 belongs to the category. On factorization, we see that 9 = 3 x 3 x 1. So 9 definitely isn't a prime number. What about 13? If you try factorization of 13, you discover that 13 = 13 x 1. That means, 13 belongs to the category.
Prime Number Theorem
Known as the 'Fundamental Theorem of Arithmetic', it states that any number (which is greater than 1), can be factorized into a product of prime numbers and this product is unique. So every composite number, is a unique product of these numbers and their powers. For example, 15 = 1 x 3 x 5, which is a unique product. Remember this theorem as it is one of the most important theorems in all of arithmetic.
So how many prime numbers are there? Mathematicians have pondered about this question for years and proved that they are indeed infinite. Here is a chart showing all those which are lesser than 1000.
Prime numbers can be looked at as building blocks of composite numbers. There are many interesting theorems and properties related to them, that you should know about.