Euclid’s proof of the infinitude of primes is justly famous. Here is one version of this proof:
Let P be a finite set of primes, and let N be the product of the numbers in P. Then N+1 is not divisible by any number in P, since it leaves a remainder of 1. But N+1 must be divisible by at least one prime, so P cannot contain all of the primes. Therefore the set of primes is infinite.
A variation of this argument shows that there are infinitely many primes of the form .
Let P be a finite set of primes of the form , and let N be the product of the primes in P. Consider the number . It is not divisible by 2 or 3, nor is it divisible by any number in P. But it is not possible that all prime factors of have the form , because the product would have the form as well.
Therefore, has at least one prime factor of the form that does not belong to P. Therefore, P cannot contain every prime of the form , so the set of primes of this form is infinite.
It takes a little bit more work to prove that there are infinitely many primes of the form . Here is a proof of that fact.
Let P be a finite set of primes of the form , and let N be a number that is divisible by every number in P. Assume that N is also divisible by 6. Let p be a prime divisor of .
Note that . so p divides , or in other words and so .
Recall that the order of N modulo p is the least positive k so that . The order must divide 6. so k = 1, 2, 3, or 6. But , so the order cannot be 1 or 3.
Can the order be 2? If and then . This would be bad, because then p would divide both and ; but , contradiction.
Thus N has order 6 mod p, and the group of units mod p has order , so 6 divides , which means that p has the form . Therefore, P does not contain all primes of the form , so the set of primes of this form is infinite.