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Prime numbers from infinite prime number proof 
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#1
Aug2213, 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? (2,3,5,7,11,13) (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. 


#2
Aug2213, 11:15 AM

P: 229

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/nonprimality 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. 


#3
Aug2213, 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 widelyused 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). 


#4
Aug2213, 12:11 PM

P: 221

Prime numbers from infinite prime number proof
Thanks for the inf



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