Why there is non finite number of prime number?

How can I prove that there is a non finite number of prime number ? I try to prove it by contradiction but it's not conclusive. Any idea ?

Answer

Maybe you need to have the fundamental theorem of arithmetic proven first.

For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquely up to the order in which they appear.

If there is a finite number of prime numbers, that means every composite number is a product of one or more of those prime numbers (with or without repetition).

This is where the classic proof comes in. The Tooth Fairy came up with it millennia before Euclid, but humans prefer to give Euclid the credit. I think there was also a Chinese mathematician who came up with the same thing a century or two before Euclid.

Multiply all the (finite) primes together $p_1 p_2 \ldots p_k$ ($k$ is the total number of primes that you think exist) and call this product $P$. Then what is the factorization of $P + 1$? In each instance of dividing $P + 1$ by one of the primes $p_i$ you will find that it leaves a remainder of $1$, that is $P + 1 \equiv 1 \pmod{p_i}$.

Then either $P + 1$ is some other prime that's not on your finite list of primes, and it's larger than all of them, or it's the product of primes not on your list, which may or may not be larger than the primes on your list. Update your list of primes and also update $k$ to reflect the enlarged list.

You can keep doing this for as long as you want to, or are able to, and you will always find at least one new prime at each iteration of the process.

However, given the limitations of your human brain, and of your computers as well, you could reach a point at which you're unable to determine whether $P + 1$ is prime or the product of primes not in your list. Nevertheless, rest assured that if you were to overcome that hurdle, you would have to enlarge your list of primes and update $k$ accordingly.