20080819, 19:34  #1 
Feb 2008
2^{5} Posts 
"Rare" Primes
I am looking for "rare" prime numbers. For purposes of this tread a prime number is rare if there are 10 or less known examples. Even if it is believed that there is an infinate number of primes of a partiocular type; it is rare if there are 10 or less known examples.
Even primes n n=2 Generalized Fermat 10^2^n+1 n=1 Subfactorial !n n=2 Perfect number 1; n is a perfect number n=6 Sequential prime of type (1234567890)n1 n=17, 56 Subfactorial +1; !n+1 n=2, 3 Type: n^n^n +1 n=1, 2 Wilson primes; (n1)!+1 is divisible by n^2 Subfactorial  1; !n1 n= 5, 15, 17 Type: n^n+1 n=1, 2, 4 Double Mersenne; 2^n1; where n is a Mersenne prime n=2, 3, 5, 7 Perfect number +1; where n is a perfect number n= 6, 28, 496, 137,438,691,328 Fermat prime; 2^n+1 n=0, 1, 2, 3, 4 Repunit containing only decimal digit 1; n= number of digits n=2, 19, 23, 317, 1,031 
20080819, 19:50  #2  
"Robert Gerbicz"
Oct 2005
Hungary
2624_{8} Posts 
Quote:


20080819, 20:40  #3 
Feb 2008
40_{8} Posts 
All primes Rare
Is everyone from Hungary a sarcastic moron?

20080819, 22:54  #5 
∂^{2}ω=0
Sep 2002
República de California
2^{2}·3·5·193 Posts 
LOL  allow me to try to inject a bit of international diplomacy with a mathematical flavor, by saying only that
"Not all sarcastic morons are from Hungary." And that is all my government [which is not Hungarian, and has no special interest in the Goulash markets] has authorized me to say on the matter. Last fiddled with by ewmayer on 20080820 at 16:28 Reason: Removed stray text caused by a cosmicraycascade of sarcastic muons 
20080819, 22:56  #6 
Feb 2008
2^{5} Posts 
Have you looked here?
Thank you for your tip!

20080820, 03:03  #7 
Feb 2006
Denmark
2×5×23 Posts 
As hinted by a not entirely moronic Hungarian, it strongly depends on which "types" you allow. http://primepuzzles.net/puzzles/puzz_225.htm has some possibilities.
It's easy to construct rare prime forms by picking a quickly growing function with one or a few early primes. You mention Generalized Fermat 10^2^n+1, but there is no base b with more than 7 known primes b^2^n+1, and finding one with more than 10 looks very hard. The record is 7 for b=2072005925466 at http://primepuzzles.net/puzzles/puzz_399.htm If you want relatively notable named forms then some candidates are at http://en.wikipedia.org/wiki/List_of_prime_numbers (look for comments like "only known"). In addition to your list of proven repunit primes, there are known probable primes for n = 49081, 86453, 109297, 270343. There is no known WallSunSun prime although infinitely many are expected to exist. 
20080820, 11:37  #8 
Nov 2003
16444_{8} Posts 
The reply to your post was accurate. "rare prime" is a poorly conceived
notion at best because as the reply shows it is TRIVIAL to construct subsets of the integers containing only finitely many primes under according to some rule. I am afraid that YOUR original question shows that you are the moron. It shows a total lack of mathematical understanding. 
20080820, 14:31  #9  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Quote:
The original question's appropriate uses of the mathematical terms "Generalized Fermat", "Subfactorial !n", "Wilson primes", "Double Mersenne", "Fermat prime", and "Repunit" are unlikely to have been composed by someone with a "total lack of mathematical understanding". Are you genuinely unable to discern, or at least politely respond to, the intent behind awkward wordings of mathematicallyrelated postings? Or is it instead a matter of using this forum to vent anger that might otherwise, and less desireably, be expressed elsewhere in your life? Last fiddled with by cheesehead on 20080820 at 14:40 

20080820, 15:18  #10  
Nov 2003
2^{2}·5·373 Posts 
Quote:
as coming from a moron. And knowing the NAME of something is not the same as understanding it. (A paraphrased quote from Richard Feynman). The fact that the O.P. knows the names of a few objects is not an indication that he understands mathematics. The original query, as posed, used vague English words (e.g. rare prime) to try to convey some mathematical idea. Mathematics is a domain of knowledge in which it is possible to state PRECISELY what is intended. The fact that the original poser used vague language and gave a very poorly posed question is what makes clear that he lacks understanding of mathematics. The first response to the problem was a totally correct and precise response to WHAT WAS ASKED. And then the O.P. labelled the response as coming from a moron. I notice that you failed to chide the O.P. for his response. Can you say "double standard"?? 

20080820, 18:14  #11  
Mar 2004
3·127 Posts 
Quote:
The probability of that szenario is still pretty small. 

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