Solution 1: There are an infinite number of prime numbers

Here is the solution to Problem 1: Prove that there are an infinite number of prime numbers.

Note:  The argument makes use of proof by contradiction.  If you are not familiar with this type of argument, a tutorial can be found here and you can watch a video here.

Claim:  There are an infinite number of prime numbers.

(1)  Assume that there are only a finite number of prime numbers.

(2)  Let $x$ be the largest such prime.  

(3)  It follows that $x!$ is divisible by all primes. 

Note:  $x!$ is the factorial for $x$.  If you are not familiar, you can learn about factorials here or see a video here.

(4)  But then, no prime divides $x! + 1$ since all such divisions will have a remainder of 1.

(5)  Thus, we have a contradiction and can reject our assumption in step 1.

Either $x!+1 > x$ is a prime itself (which contradicts our assumption in step 1 that $x$ is the largest prime number) or $x!+1$ is composite and divisible by a prime number greater than $x$ (which also contradicts this same assumption in step 1).