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

 

[CellCoree]

Does this mean there are infinitely many primes in between?

I find this thought experiment interesting and thought-provoking. The proof that there are infinitely many primes is a fundamental result in mathematics, and it has been studied and proven by many mathematicians throughout history. It is a powerful and elegant proof that relies on the fundamental properties of prime numbers and their relationship to each other.

To answer your questions, prime numbers of the form (product of primes) +- 1 do not have a specific name, but they are often referred to as "near-primes" or "almost-primes." These numbers have been studied extensively and have applications in cryptography and number theory.

As for your second question, it is still an open problem whether there are infinitely many near-primes. Some mathematicians believe that there are infinitely many, while others believe that there are only finitely many. This is still an active area of research and has not been proven definitively.

Your thought experiment raises an interesting question about the distribution of primes and whether there are infinitely many primes in between these near-primes. This is also an open problem in mathematics and has been studied by many mathematicians. It is a complex problem that has not been fully understood yet, but it is an exciting area of research.

In conclusion, the concept of near-primes is a fascinating one that has been studied and explored by mathematicians. Your thought experiment adds an interesting perspective to this topic and highlights the ongoing research in this area. It is a reminder that there is still much to discover and understand about the mysterious and beautiful world of prime numbers.