Around 300 B.C. Euclid of Alexandria proved that there are infinitely man prime numbers.

He proved it by contradiction which is one of well-known proof methods.

Suppose that prime numbers are finite and the largest prime number is p. The complete list of prime numbers becomes: 2, 3, 5, 7, ..., p

Consider a number obtained by muliplying all prime number and adding 1.
N = 2 x 3 x 7 x ... x p + 1

The number N is certainly greater than p and it cannot be a prime number because p is the greatest prime number. Hence N must be divisible by at least one prime number.

If you divide N by any number from the list, you always have a remainder of 1. That means N cannot be divisible by any number from the list.

This gives rise to the contradiction and hence the original hypothesis is not correct. Therefore, there are infinitely man prime numbers.