"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 
[QUOTE=Housemouse;139515]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.
[/QUOTE] Every prime is rare: p is a prime then p is rare because it is only the solution of the equation xp=0. 
All primes Rare
Is everyone from Hungary a sarcastic moron?

Have you tried looking [URL="http://primes.utm.edu/top20/home.php"]here[/URL]?
BTW: I'm from Poland :wink: 
[QUOTE=Housemouse;139523]Is everyone from Hungary a sarcastic moron?[/QUOTE]
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. 
Have you looked here?
Thank you for your tip!

As hinted by a not entirely moronic Hungarian, it strongly depends on which "types" you allow. [url]http://primepuzzles.net/puzzles/puzz_225.htm[/url] 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 [I]more[/I] 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 [url]http://primepuzzles.net/puzzles/puzz_399.htm[/url] If you want relatively notable named forms then some candidates are at [url]http://en.wikipedia.org/wiki/List_of_prime_numbers[/url] (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. 
[QUOTE=Housemouse;139523]Is everyone from Hungary a sarcastic moron?[/QUOTE]
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 [b]you [/b] are the moron. It shows a total lack of mathematical understanding. 
[quote=R.D. Silverman;139544]I am afraid that YOUR original question shows that [B]you [/B]are the moron. It shows a total lack of mathematical understanding.[/quote]Perhaps your fear has clouded your reasoning, professor.
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? 
[QUOTE=cheesehead;139550]Perhaps your fear has clouded your reasoning, professor.
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?[/QUOTE] I am not the one who labelled the response to the original question 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"?? 
[QUOTE=Jens K Andersen;139532]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 [I]more[/I] 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 [url]http://primepuzzles.net/puzzles/puzz_399.htm[/url]
[/QUOTE] Maybe there is even a Generalized Fermat with more than 7 primes in the range b<10^15 or so. The sequence above assumes that n=0..6 of b^2^n+1 is prime. Maybe there is a generalized Fermat which has more than 7 primes that are not consecutive. For example if n=0,1,2,3,4,5,7,8 is prime. The probability of that szenario is still pretty small. 
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