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Prime numbers from infinite prime number proof

by jfizzix
Tags: infinite, number, numbers, prime, proof
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Aug22-13, 10:52 AM
P: 221
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) +-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.
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Aug22-13, 11:15 AM
P: 248
I don't know a name of primes of the form [itex]\pm1+\prod_{p\in P} p [/itex] for [itex]P[/itex] 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 [itex]P[/itex] can be chosen to be the finite set of all primes.
Aug22-13, 11:21 AM
P: 160
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).

Aug22-13, 12:11 PM
P: 221
Prime numbers from infinite prime number proof

Thanks for the inf

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