There are

infinitelymany prime numbers..

**Godefroy Harold Hardy** (1877-1947) english mathematician, collected in his book self-justification of a mathematical two famous theorems of classical Greek mathematics. The first one states the primes are infinite:

We must to **prove** that there are infinitely many prime numbers, ie, the set of numbers:**2,3,5,7,11,13,17,19,23, …**

has **infinite elements**. To do this **we assume** that this set has a finite number of elements, that is, there is a last number P.

On this hypothesis we define the number **Q** how:**Q=2·3·5·7·….·(P+1)**

We know that if **P** is the last prime number, the number **P + 1** is not divisible by any number since by assumption is **not** a prime number, so it is not divisible by 2,3,5,7 and P … it is only divisible by **P+1** and 1 and so this we deduce that **Q is a prime number** but being **Q** a larger than **P**, we arrive at a **contradiction** since the hypothesis was that the largest prime number is **P**.

Consequently the hypothesis is **false** and therefore all the set of **primes** has **infinite elements**.