# Are prime numbers infinite?

Apr 4, 2016

There are an infinite number of prime numbers.

#### Explanation:

To prove this for yourself, suppose you only know a finite number of prime numbers - call them ${p}_{1} , {p}_{2} , \ldots , {p}_{n}$

Consider the number $q = \left({p}_{1} \times {p}_{2} \times \ldots \times {p}_{n}\right) + 1$

$q$ is not divisible by any of the prime numbers you know, so it is either prime itself or there is some other prime ${p}_{n + 1}$ which is a factor of $q$.

In either case, your list of primes was incomplete.

So the number of primes is unbounded.