SOLUTION: Prove that there are infinitely many primes of the form 6n − 1.

Suppose there are finitely many primes of the form 6n − 1
and these are p1, p2, ..., . 

Take M = . 

If M is a prime, we have a contradiction, because, M is of the form 6n - 1 but not on our list. 

If M is not a prime, then it has some prime factors , none of which
are 2, 3, p1, p2, . . . , ,  so they must be of the form  6n + 1  or  6n − 1. 
But if all the    are of the form  6n + 1 then their product would also have this form
which M does not. Therefore,  at least one of the    is a new prime of the form 6n-1. 

Thus our set was not complete, and we got a contradiction with the original assumption.

So, there are in fact infinitely many primes of this form.