Prime numbers from infinite prime number proof

[jfizzix]

I imagine most everyone here's familiar with the proof that there's an infinite number of primes:

If there were a largest prime
you could take the product of all prime factors
add (or take away) 1 and get another large prime (a contradiction)

So what if you search for larger primes this way?

(2,3,5,7,11,13)

(2*3) +-1 = 6 +-1 = {5,7}
(2*3*5) +-1 = 30+-1 = {29.31}
(2*3*5*7)+-1 = 210+-1 = {209,211} (209 is not prime)
(2*3*5*7*11)+-1 = 2310+-1 = {2309,2311}
(2*3*5*7*11*13)+-1 = 30030+-1={30029,30031} (30031 is not prime)

I have two questions:
Do prime numbers of this sort have a special name? (like Marsenne primes are (powers of 2) +-1?)
Are there infinitely many of them?

This was just an odd thought I had. You can keep going and find products where neither one above or one below is a prime.

 

[economicsnerd]

I don't know a name of primes of the form \pm1+\prod_{p\in P} p for P a finite set of primes.

One comment, though. I'm not sure whether primality/non-primality of numbers of the above form is that interesting ("interesting" being too subjective for my comment to make any sense :P). The argument to which you're referring generates primes like that based on a hypothesis we know to be false: namely, that P can be chosen to be the finite set of all primes.

 

[eigenperson]

The products of the first n primes are called the primorials.

If you add 1 to these, you get the Euclid numbers.

If you subtract 1 instead, you get the Kummer numbers.

The prime Euclid numbers (or prime Kummer numbers) don't have special names. They are just the "prime Euclid numbers." I guess you could call them "Euclid primes" (or "Kummer primes") if you wanted to be fancy, but this is not widely-used terminology. You can find a list of the first few prime Euclid numbers on OEIS. I believe the question of whether this list goes on forever is unsolved.

As far as I know, the combined list of prime Euclid numbers and prime Kummer numbers has no name (and isn't even on OEIS as far as I can tell).

 

[jfizzix]

Thanks for the inf