# Prime numbers

What is a prime number?

In mathematics, we say a natural number $p$ is a prime number if p has exactly 2 factors. For example, 11 is a prime number, because the only factors are 1 and 11. It is easy to see that the only prime numbers less than 10 are 2, 3, 5 and 7. The smallest prime number is 2. The only even prime number is also 2. Another equivalent definition of prime number is a natural number greater than 1 that is not a product of two natural numbers greater than 1. For example, 10 is not a prime number, because $10=2\times 5$.

What is a composite number?

If a number greater than 1 is not prime, we call it a composite number. The smallest composite number is 4.

Is 1 a prime number?

Note than 1 is not a prime number and it is not a composite number.

List of prime numbers 1 to 100:

In total we have 25 prime numbers less than 100. Here is the list:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

There are a total of 168 prime numbers less than 1000.

How many prime numbers are there?

There are infinitely many prime numbers. There are many proofs for this statement. Let’s prove it here with the most famous method:

On the contrary, let’s assume we have finitely many prime numbers, let’s call them $p_1,…,p_k$. Let $n=p_1\times p_2\times … p_k +1$, it is not difficult to see than $n$ is greater than all $p_{1},…,p_{k}$, so $n$ is not one of them, and then, $n$ is a composite number. So $n$ has a prime factor, let’s say $p_i$ is a factor of $n$. It means than $\dfrac{n}{p_i}$ is an integer, but $\dfrac{n}{p_i}=\dfrac{ p_1\times p_2\times … p_k }{p_i} +\dfrac{1}{p_i}=p_1\times p_2\times p_{i-1}\times p_{i+1}\times … p_k +\dfrac{1}{p_i}$, which is not an integers, that contradicts our assumption that we have finitely many prime numbers. So, it proves than There are infinitely many prime numbers.

How to check if a number is prime?

To see if a number $n>2$ is a prime number first we check if $n$ is divisible by 2, if it is divisible by 2, then is it not a prime number, then we check divisibility by 3, and then by 5,… . We don’t need to continue to $n$. It is enough to check until $\sqrt{n}$.

For example, to check the number 53, we check this test until $\sqrt{59}$. So we only need to check this test for number, 2, 3, 5 and 7.

What is prime factorization?

Prime Factorization” is factoring our number $n$ into prime factors (prime numbers multiply together to make $n$. For example, the prime factorization of 100 is $100=2^{2}\times 5^{2}$. Prime factors of a number $n$ are prime numbers that are multiplied together to get $n$. As an example, 2 and 5 are prime factors are 10. Note that Prime Factorization is unique.