"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. 
[QUOTE=R.D. Silverman;139555]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.[/QUOTE] Are you implying that anybody asking for "interesting" counterexamples, an "elegant" proof, or "instructive" examples doesn't understand math, because the quoted words don't have a precise mathematical definition? 
[quote=R.D. Silverman;139555]I am not the one who labelled the response to the original question as coming from a moron.[/quote]... nor did I say you were. As I quoted, you wrote "I am afraid that YOUR original question shows that [B]you [/B]are the moron." I.e., you labelled the original poster as a moron, and your posting is what I responded to.
[quote]And knowing the NAME of something is not the same as understanding it. (A paraphrased quote from Richard Feynman).[/quote]Lewis Carroll wrote something similar. [quote]The fact that the O.P. knows the names of a few objects is not an indication that he understands mathematics.[/quote]No, but he not only knew the names, but also used them appropriately, and it was that _combination_, not merely knowing names, that I considered unlikely to come from someone lacking understanding: "The original question's appropriate uses of the mathematical terms ... are unlikely to have been composed by someone with a 'total lack of mathematical understanding'." [quote]The original query, as posed, used vague English words (e.g. rare prime) to try to convey some mathematical idea.[/quote]So? This is a public online forum, not a professional journal or one of your classes. [quote]Mathematics is a domain of knowledge in which it is possible to state PRECISELY what is intended.[/quote]So? This thread's posting content clearly involves ideas outside that domain as well as inside it. And if you're requiring that all posts achieve some ideal of precision even outside of mathematics, then for consistency you need to disqualify a whole bunch (pardon the imprecision) of your own words. [quote]The first response to the problem was a totally correct and precise response to WHAT WAS ASKED.[/quote]So? i don't dispute that. [quote]And then the O.P. labelled the response as coming from a moron.[/quote]... and then you labelled the O.P. as a moron, which is not the first (or tenth, or twentieth, ...) time you've done that (i.e., labelling a poster as a moron) in forum threads. [quote]I notice that you failed to chide the O.P. for his response.[/quote]You already did that, it seemed to me. I didn't see any use in piling on. [quote]Can you say "double standard"??[/quote]When the O.P. has labelled other posters as "moron" without sufficient justification [I]as often as you have[/I] in this forum, I'll treat him by the same, single, standard. I didn't complain the first several times you did it. 
:flamewar:
This warning should be put on every one of RD Silverman's posts...because any time he posts, it's just to insult someone who might not know as much as him or used improper language or something, to which somebody nicer tries to defend the insultee. The topic is always forgotten after RD Silverman posts, largely because he ignores it and starts a flame war based on something stupid. 
[QUOTE=MiniGeek;139592]:flamewar:
This warning should be put on every one of RD Silverman's posts...because any time he posts, it's just to insult someone who might not know as much as him or used improper language or something, to which somebody nicer tries to defend the insultee. The topic is always forgotten after RD Silverman posts, largely because he ignores it and starts a flame war based on something stupid.[/QUOTE] I am about to pour gas on the fire. I don't think that you are a moron. I think that you are an idiot trying to become a moron. I case you did not notice (and clearly you did not), I was [b]DEFENDING[/b] someone who the O.P. called a moron. You clearly can not read. 
[QUOTE=R.D. Silverman;139544]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] It is also TRIVIAL NOT to construct subsets of the integers containing only finitely many primes according to some rule.How can you tell the difference in ALL cases.(I am assuming you haven't hidden away a proof that there are only finitely many Mersenne Primes somewhere for starters.) 
[QUOTE=Visu;139620]It is also TRIVIAL NOT to construct subsets of the integers containing only finitely many primes according to some rule.How can you tell the difference in ALL cases.(I am assuming you haven't hidden away a proof that there are only finitely many Mersenne Primes somewhere for starters.)[/QUOTE]
Oh? We know that the are infinitely many primes in Z. We know that there are infinitely many primes in (some) A.P.'s. Most (sufficiently) fastgrowing sequence will contain finitely many primes. It is known that (with suitable definition of measure) almost all sequences A_1, A_2, A_3, ... for which sum(1/log(A_i)) converges contain finitely many primes (almost all means Lebesgue measure 1 among the uncountable set of all such sequences; It is only the uncountability of the sequences that makes this proof at all tricky.) Constructing a sequence of integers that grows faster than linear that contains infinitely many primes is NOT a trivial problem. We do have proofs for some. For example, there exists a positive real number theta such that floor(theta^3^n) is prime i.o. I know of no equivalent proof that the almost all sequences for which the above sum diverges contains infinitely many primes. (I don't even know if it has been considered; I just gave myself a research problem; but the proof MAY turn out to be simple) While it is virtually certain that the sequence (a^n 1)/(a1) for n=2,3,5,... contains infinitely primes for positive appropriate values of a (e.g. non powers) , we have no proof. Most FASTER GROWING sequences will not contain infinitely many. 
[QUOTE=MiniGeek;139592]:flamewar:
This warning should be put on every one of RD Silverman's posts...because any time he posts, it's just to insult someone who might not know as much as him or used improper language or something, to which somebody nicer tries to defend the insultee. The topic is always forgotten after RD Silverman posts, largely because he ignores it and starts a flame war based on something stupid.[/QUOTE] It was [b]not[/b] I who started the flamewar. It was the [b]O.P. [/b] If one wants to discuss mathematics, then either use proper language or don't post. Failure to use proper language shows that either you have not done proper backgound preparation or else lack the ability to discuss the problem intelligently. In both cases you should not post! I have been told repeatedly that this is not a classroom and that I am not here to teach. If this is NOT a classroom, then it becomes a (purported) discussion among peers. In that case flames are appropriate for poorly posed questions or for posts that clearly show inadequate ability/knowledge/preparation for intelligent discussion. I [b]propose[/b] we separate the math subforum into two groups: (1) A discussion of mathematics at the undergrad or higher level, with a requirement that anyone initiating a thread has thoroughly mastered high school level math (basically precalculus). I would also assume sufficient mathematical maturity to present problems that make sense. I would also presume that anyone posing a question involving math not ordinarily seen in high school such as abstract algebra, number theory, etc. has, AT A MINUMUM at least read a book on the subject and worked some of the problems. This is basically a subforum for the mathematically competent. It can even be refereed. It is the equivalent of sci.math.research, except at a lower level. (2) Questions, ideas, conjectures and general nonsense posed by newbies, and those less knowledgeable. This would include those who can not rigorously formulate their questions. I fully promise not to even reply to this subgroup. It is the equivalent of 'alt.algebra.help' However, if newbies want to play with the big boys, they need to show that they have earned the right to do so by showing that they have at least read the literature before posing any questions, and that they have formulated their problem using standard mathematical notation and terminology. There is a discussion currently in sci.math in which others (not I) have made the statement: "You have not earned the right to take up (or waste) my time" in reply to a post. John Baez created a score card for measuring cranks in general. Some of his rules are quite relevant here. I will propose some additional scores herein. (1) 10 points for trying to discuss any subject for which you have not taken a course, or read at least one book. (2) 15 points for not using standard mathematical terminology. (3) 15 points for failing to define your variables and their domain. (4) 20 points for trying to generate a discussion instead of asking a question when it is clear that you do not understand what you are trying to discuss. (5) 25 points for failing to do a web or literature search before posing an idea or question. (6) 35 points for elementary mistakes in high school level mathematics. (7) 50 points for trying to invent new mathematical terminology. (8) 50 points for trying to "reinvent the wheel". An extra 10 points for reinventing a "square wheel" (e.g. a 'new' algorithm that performs more poorly than existing ones) (9) 50 points for posing poorly defined problems, or for posing problems which show a lack of BASIC understanding of elementary aspects of the subject you are trying to discuss. (10) 100 points for both trying to invent new terminology and failing at the same time to rigorously define what that terminology really means. (11) 200 points for even trying to pose a solution to a wellstudied problem in which you are not an expert. (12) 500 points for trying to claim that knowledge of the stateoftheart "gets in the way" of creativity. (13) 1000 points for any comparison of yourself to any well known mathematician, or for trying to point out that some prior mathematican worked in some area in which he/she was not trained as if this were an excuse for your doing the same.  Note that there is a difference between asking "is the following solution correct", and asserting that you have a solution. I, in turn, will promise not to flame ANYONE unless their crank score exceeds 25 (including both my scores and those of John Baez). If you want to discuss math at the undergrad level or higher, you must earn the right to do so by demonstrating that you have the background to do so intelligently. Otherwise, you WILL get flamed. Above all, this means that any questions presented need to be rigorous in their meaning. 
[QUOTE=R.D. Silverman;139622]Oh? We know that the are infinitely many primes in Z. We know that
there are infinitely many primes in (some) A.P.'s. Most (sufficiently) fastgrowing sequence will contain finitely many primes. It is known that (with suitable definition of measure) almost all sequences A_1, A_2, A_3, ... for which sum(1/log(A_i)) converges contain finitely many primes (almost all means Lebesgue measure 1 among the uncountable set of all such sequences; It is only the uncountability of the sequences that makes this proof at all tricky.) Constructing a sequence of integers that grows faster than linear that contains infinitely many primes is NOT a trivial problem. We do have proofs for some. For example, there exists a positive real number theta such that floor(theta^3^n) is prime i.o. I know of no equivalent proof that the almost all sequences for which the above sum diverges contains infinitely many primes. (I don't even know if it has been considered; I just gave myself a research problem; but the proof MAY turn out to be simple) While it is virtually certain that the sequence (a^n 1)/(a1) for n=2,3,5,... contains infinitely primes for positive appropriate values of a (e.g. non powers) , we have no proof. Most FASTER GROWING sequences will not contain infinitely many.[/QUOTE] Strange that someone who demands precision uses terms such as "most(sufficiently)fastgrowing" , "Most FASTER GROWING" and "virtually certain" to name a few. 
[quote=R.D. Silverman;139626]I have been told repeatedly that this is not a classroom and that I am
not here to teach. If this is NOT a classroom, then it becomes a (purported) discussion among peers.[/quote]... or  if this is not a classroom then it could, alternatively, be (and, in fact, is) a discussion among notnecessarilypeers of varying abilities and backgrounds  a public forum with no entrance requirement or threshold. [quote]In that case flames are appropriate[/quote]... but in [I]other[/I] cases, they're not. [quote]I [B]propose[/B] we separate the math subforum into two groups:[/quote]A simpler alternative: create a subforum titled, "R. D. Silverman Answers ClearlyPhrased Questions Asked by Knowledgable Persons", or something similar specifying the requirement right in the title, to dissuade newbies from stumbling in by accident. 
[QUOTE=cheesehead;139628]... or  if this is not a classroom then it could, alternatively, be (and, in fact, is) a discussion among notnecessarilypeers of varying abilities and backgrounds  a public forum with no entrance requirement or threshold.
... but in [I]other[/I] cases, they're not. A simpler alternative: create a subforum titled, "R. D. Silverman Answers ClearlyPhrased Questions Asked by Knowledgable Persons", or something similar specifying the requirement right in the title, to dissuade newbies from stumbling in by accident.[/QUOTE] I thought we already had a Miscellaneous Math Threads. Maybe all posts should go there by default and get voted into the main Math thread (by the moderators?) if enough people think the post has its merits.This will dissuade Dr Silverman from stumbling in by accident. 
[QUOTE=Visu;139627]Strange that someone who demands precision uses terms such as "most(sufficiently)fastgrowing" , "Most FASTER GROWING" and "virtually certain" to name a few.[/QUOTE]
These are precise terms in mathematics. Go learn why. 
[QUOTE=Visu;139629]I thought we already had a Miscellaneous Math Threads. Maybe all posts should go there by default and get voted into the main Math thread (by the moderators?) if enough people think the post has its merits.This will dissuade Dr Silverman from stumbling in by accident.[/QUOTE]Not obviously a bad idea.
Are there enough moderators willing to put in the effort? I'd help out and Bob certainly should! He's often shown himself willing to help the intelligent but uneducated. If he doesn't flame anything in the MMT and moves out of there anything he considers worthwhile, that sounds like a valuable service to me. On the other hand, flame wars do provide entertainment for bystanders who are mature enough not to be fazed by them and trolling is a wellestablished passtime, one in which I indulge myself every now and again including here. :devil: Paul 
[QUOTE=xilman;139634]Not obviously a bad idea.
Are there enough moderators willing to put in the effort? [/QUOTE] Well there are moderators indulging in changing the subject heading here [url]http://www.mersenneforum.org/showthread.php?t=10429[/url] and elsewhere. This should keep them gainfully occupied. (Maybe it is less fun but...) [QUOTE=xilman;139634] I'd help out and Bob certainly should! He's often shown himself willing to help the intelligent but uneducated. If he doesn't flame anything in the MMT and moves out of there anything he considers worthwhile, that sounds like a valuable service to me. [/QUOTE] I am not too sure how many posts will get moved out:smile: [QUOTE=xilman;139634] On the other hand, flame wars do provide entertainment for bystanders who are mature enough not to be fazed by them and trolling is a wellestablished passtime, one in which I indulge myself every now and again including here. :devil: Paul[/QUOTE] To paraphrase Ltc Kilgore ,"I too love the smell of napalm in the morning". 
[QUOTE=Visu;139627]Strange that someone who demands precision uses terms such as "most(sufficiently)fastgrowing" , "Most FASTER GROWING" and "virtually certain" to name a few.[/QUOTE]
Hey moron! Go learn some mathematics. I *defined* what sufficiently fast growing meant: i.e. sum (1/log(A_i)) *converges*. 'most' is certainly well defined as is 'virtually certain': go learn some measure theory. Learn what it means to be a set with asymptotic density 0. Learn the precise meaning of 'occurs with probability 1'. 
[QUOTE=cheesehead;139628]... or  if this is not a classroom then it could, alternatively, be (and, in fact, is) a discussion among notnecessarilypeers of varying abilities and backgrounds  a public forum with no entrance requirement or threshold.
[/QUOTE] Wonderful. A forum where the ignorant try to discuss subjects with the incompetent. It is fair for competent people to ask that such a forum be labelled as such so that we don't waste our time. There is no way to know, a priori, whether a post contains meaningful content until after one has read it and digested its intent. By then, of course, one has already wasted the effort..... I think creating such a forum is a terrific idea! But as I proposed, you should create a separate forum for competent discussion. If you want the math subforum to remain 'one size fits all', then you can continue to expect flames. 
[QUOTE=R.D. Silverman;139626]
<list deleted> [/QUOTE] Allow me to add: (11a) 100 points for posting a 'conjecture' in which it is clear that you have not bothered to test it yourself via numerical example. Especially if trivial counterexamples are available. 
[quote=R.D. Silverman;139643]
If you want the math subforum to remain 'one size fits all', then you can continue to expect flames.[/quote] As has already been pointed out, there is already a "Miscellaneous Math" forum. This "Math" forum is under the major heading of "GIMPS". "Case Law" needs to be applied to arrive at what the boundaries are in practice. For what it's worth, I think the OP's "Are all Hungarians sarcastic morons?" was contrary to the good spirit that makes the whole forum enjoyable. However justified RDS's retort was, it was simply typical of the man. David 
I can't wait for the competition to see who can score the most on the RDS crankometer.
Is it possible to score maximum with just one posting? Is it possible to score multiple times for one list item in one posting (thus making any maximum potentially infinite)? 
[quote=R.D. Silverman;139544] I am afraid that YOUR original
question shows that [B]you [/B] are the moron[/quote] [quote=MiniGeek;139592]any time he posts, it's just to insult someone[/quote] [quote=R.D. Silverman;139618]I am about to pour gas on the fire. I don't think that you are a moron. I think that you are an idiot trying to become a moron. I case you did not notice (and clearly you did not), I was [B]DEFENDING[/B] someone who the O.P. called a moron. You clearly can not read.[/quote] I never said you called the OP a moron, I said you only post to insult someone, such as both of your above posts I quoted. And yes, actually, you did call the OP a moron, in case you forgot already. Let's look at the timeline of events:[LIST=1][*]OP posts about rare primes[*]Hungarian says every prime is rare by some definiton[*]OP says Hungarian is a sarcastic moron[*][other discussion][*]You say OP is a moron for calling the Hungarian a sarcastic moron[/LIST]See? Scroll up to refresh your memory with the precise wordings, if you wish... 
[QUOTE=MiniGeek;139651]I never said you called the OP a moron, I said you only post to insult someone, ..[/QUOTE]
They insult themselves everytime they mimic the village idiot by spouting ignorant gibberish in a public forum. I only identify them as such. 
"Other discussion" being a helpful link, thankfully greeted by OP
and Ernst's attempt at dousing the flames. Then a very conciliatory and constructive post from Jens. Now enter RDS with flame gun blazing. 
[QUOTE=davieddy;139654]"Other discussion" being a helpful link, thankfully greeted by OP
and Ernst's attempt at dousing the flames. Then a very conciliatory and constructive post from Jens. Now enter RDS with flame gun blazing.[/QUOTE][b]Ok, everybody. I enjoy a flame war at least as much as anyone else but this one has gone on long enough.[/b] Either drop the "moron"slinging and the "you did it first  no you did" ping pong or I'll lock the thread. We've already had a concrete suggestion about how to improve the environment for all concerned. Paul 
[QUOTE=R.D. Silverman;139641]Hey moron! Go learn some mathematics. I *defined* what sufficiently
fast growing meant: i.e. sum (1/log(A_i)) *converges*. 'most' is certainly well defined as is 'virtually certain': go learn some measure theory. [/QUOTE] I did not question your lack of definitions. I questioned your use of "most" and "virtually certain" as qualifiers. Why didn't you say "all" instead of "most" and "certainly" instead of "virtually certain"? The OPs original query may have been poorly phrased but it does have its merits. For example while there are only 3 known Wilson Primes it has been conjectured that infinitely many exist. He only seemed to be asking if he has missed any from his list. [QUOTE=R.D. Silverman;139641] Learn what it means to be a set with asymptotic density 0. Learn the precise meaning of 'occurs with probability 1'.[/QUOTE] A finite set HAS to have an asymptotic density 0, but having an asymptotic density 0 does not mean that the set is finite. I am not certain where the 'occurs with probability 1' comes in. 
[QUOTE=Visu;139666]I did not question your lack of definitions. I questioned your use of "most" and "virtually certain" as qualifiers. Why didn't you say "all" instead of "most" and "certainly" instead of "virtually certain"?
The OPs original query may have been poorly phrased but it does have its merits. For example while there are only 3 known Wilson Primes it has been conjectured that infinitely many exist. He only seemed to be asking if he has missed any from his list. A finite set HAS to have an asymptotic density 0, but having an asymptotic density 0 does not mean that the set is finite. I am not certain where the 'occurs with probability 1' comes in.[/QUOTE] Sigh. You need to read my signature. It pertains to you. I did not say "all", because it would be incorrect. A subset (of an infinite set) with density 1 need not contain ALL elements. I did not say "certain" because it too would be incorrect. A probability subspace can have measure 1, yet not contain ALL elements of its parent. And where in hell did I ever say that a set has to be finite to have density 0??? And your ignorance clearly shows in your lack of understanding of the relevance of "probability 1". Once again, GO STUDY SOME MEASURE THEORY. 
Sorry
I am sorry that I overreacted to the first reply to my post, I mistakenly thought he was being sarcastic.
I did not intend to start a flame war. I am sorry I lack the mathemtical knowledge RDS demands of everyone. Although I lack mathematical knowledge, I am very curious about prime numbers. I will try to improve my future posts. RDS, please do not waste your time reading my future posts; they will probably not meet your criteria. If you do not read them; we both will be happier. Thank you 
Well said Housemouse.

[quote=R.D. Silverman;139643]It is fair for competent people to ask that such a forum be labelled as such so that we don't waste our time.[/quote]You have had [U]years[/U] in which to learn that this forum has postings by those you deem incompetent and ignorant, yet you still lack the selfcontrol to avoid wasting your time by responding to questions by those you deem incompetent.
If you were in full control of yourself, you could refrain from responding to questions you deem inadequate to meet the standards you espouse. I've pointed out to you that there are other participants who are willing to take the time and effort to respond to those questions in a different manner, so you need not fear that anyone's questions will go unanswered. That you continue to pester folks here in the way you do is a sign that you either have an uncontrolled emotional compulsion to respond, and/or that you use this forum to express anger that would otherwise leak out in other areas of your life. [quote]There is no way to know, a priori, whether a post contains meaningful content until after one has read it and digested its intent. By then, of course, one has already wasted the effort.....[/quote][U]But at that time you could still choose not to waste further time and effort, by the simple choice of not composing and posting a response.[/U] You're certainly capable of figuring this out, yet you continue to waste that further time and effort by posting your responses to those. Clearly, then, you are not making a rational decision, but are driven by irrational forces beyond your conscious control. [quote]If you want the math subforum to remain 'one size fits all', then you can continue to expect flames.[/quote]... from people who cannot control themselves according to their own publiclyexpressed criteria! 
[QUOTE=xilman;139634]Not obviously a bad idea.
Are there enough moderators willing to put in the effort? I'd help out and Bob certainly should! He's often shown himself willing to help the intelligent but uneducated. If he doesn't flame anything in the MMT and moves out of there anything he considers worthwhile, that sounds like a valuable service to me. On the other hand, flame wars do provide entertainment for bystanders who are mature enough not to be fazed by them and trolling is a wellestablished passtime, one in which I indulge myself every now and again including here. :devil: Paul[/QUOTE] I don't do much moderating in Misc. Math (I haven't seen things get out of hand, and threads that I think should get moved out usually get snapped up by another mod before I get to them) and I'd be willing to filter through messages and kick them over here if they seem worthy 
Oh dear, this has become a sad thread.
I usually score quite highly (crankometer) on many of my questions I pose to this group. I persevere because I quite like recreational maths, and I am always grateful for helpful responses. I like to be enlightened. And some responders are better than others in bringing out the best in me. But I persevere also here because I have a relatively thick skin. I duck and weave the unpleasant. My preference is to receive thoughtful responses that both educate and encourage. I think the forum already has two types of sub groups, and threads usually belong to either one or the other, so there is no real need to change. This is a lively group and there are some great and helpful minds out there. It would be good to keep it that way. 
Well, it's nice to know I'm not the only one who starts flame wars. :smile:
I have an idea. Why doesn't everyone start responding to the question that was originally posed. Even if it's not worded quite the way it is needed for precise mathematics, to me it's clear what the O.P. intended: Find prime forms with few primes but be reasonable about it by not allowing such outlandish forms that they become large very quickly. In other words, don't make it some stupid form that gets large so quickly that while it cannot be proven to never have a prime, it has such a miniscule chance that it is uninteresting for mathematical discussion. RDS, I realize this is still very vague. I'm only stating what it APPEARS that the O.P. intended. I claim no understanding of math higher than high school calculus and freshman level algebra. This appears to be an interesting topic to expound upon. Let's set up some parameters and rules about what constitue an 'interesting rare prime form', i.e. not something stupid like k*2^(n^n^n^n^n^n)1. Then go from there. The O.P. gave us a starting point with some interesting forms. We just need to frame the parameters for 'rare primes' of other forms. Gary 
[QUOTE=gd_barnes;140258]Well, it's nice to know I'm not the only one who starts flame wars. :smile:
I have an idea. Why doesn't everyone start responding to the question that was originally posed. Even if it's not worded quite the way it is needed for precise mathematics, to me it's clear what the O.P. intended: Find prime forms with few primes but be reasonable about it by not allowing such outlandish forms that they become large very quickly. In other words, don't make it some stupid form that gets large so quickly that while it cannot be proven to never have a prime, it has such a miniscule chance that it is uninteresting for mathematical discussion.Gary[/QUOTE] Sigh. You need to read my signature. It applies to you. No matter how many times I say it, the message does not appear to get through to some people. It does not take knowledge of ANY advanced mathematics to see that you are spouting VAGUE GIBBERRISH. It isn't math, it is NONSENSE. Without a definition of the terminology involved, what you say is MEANINGLESS. What does "be reasonable about it" mean? What does "outlandish" mean? What does "stupid form" mean? What does "uninteresting" mean in this context. And who is the audience? Wannabees? Cranks? Or real mathematicians??? The O.P. did not ask a question that CAN be answered in any meaningful way. 
I am not a mathematician. I only had 3 years of high school math. But I am curious. I defined what I meant by "rare" for the purpose of this thread.
Is it so easy, as to be trival, to construct formulas that you can prove will result in exactly 1 prime, 2 primes etc. up to 10? 
[QUOTE=Housemouse;140285]Is it so easy, as to be trival, to construct formulas that you can prove will result in exactly 1 prime, 2 primes etc. up to 10?[/QUOTE]
Yes! It is trivial to create a function that will generate any finite number of values. By selecting a set of prime values to be generated, the resulting formula will meet your criterion. 
Do these functions have an infinite number of values but a specific number of prime values?

[QUOTE=Housemouse;140289]Do these functions have an infinite number of values but a specific number of prime values?[/QUOTE]
Go look up the definition of 'domain' and 'range' in the context of studying functions. 
At this point, I will join Dr. Silverman and suggest that you study the elementary mathematics related to the definitions and properties of the terms that you are attempting to use. With some more of that understanding, you might realize how your question is just ridiculous.
Suggested topics: Domain, range, mapping, function (Bob: Sorry, I didn't realize that you were formulating the same sort of reply) 
Wacky
Can you please give me an example of one function that has an infinate number of values, but can be proven to have exactly 10 prime values?

[QUOTE=Housemouse;140298]Can you please give me an example of one function that has an infinate number of values, but can be proven to have exactly 10 prime values?[/QUOTE]
f(x) = x, for x = 2,3,5,7,11,13,17,19,23,29 = x^2 for all other x. 
[QUOTE=Housemouse;140298]Can you please give me an example of one function that has an infinate number of values, but can be proven to have exactly 10 prime values?[/QUOTE]Perhaps: f(x)=29x^2 ?

[QUOTE=Housemouse;140298]Can you please give me an example of one function that has an infinate number of values, but can be proven to have exactly 10 prime values?[/QUOTE]I can.
[spoiler]Let f(x) be the function such that f(x) = x for 1<=x<=29 and f(x) = 4x for all all other values of x.[/spoiler] Paul 
Housemouse,
Yes, I can. However I choose to not do so because, as noted previously, it is trivial. If, instead, you will show that you have done the "homework" that I have suggested, and still cannot formulate such a function, I will be happy to continue the discussion. 
[QUOTE=retina;140300]Perhaps: y=29x^2 ?[/QUOTE]
No. 
[QUOTE=R.D. Silverman;140304]No.[/QUOTE]f(x)=29x^2 ?

[QUOTE=retina;140305]f(x)=29x^2 ?[/QUOTE]
Although not explicitly stated, I believe that the domain is N. Now, f(x) is prime for x = 0, 4 and no other. If you accept the more general definition of prime (i.e. not restricted to just N) then f(x) will be prime i.o. (although a proof is lacking). If we allow x \in R, then f(x) is indeed prime the required number of times. 
[quote=R.D. Silverman;140306]Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general definition of prime (i.e. not restricted to just N) then f(x) will be prime i.o. (although a proof is lacking). If we allow x \in R, then f(x) is indeed prime the required number of times.[/quote] If we allow real numbers as x, then f(x) is prime 19 times  9 times for negative x values as f(x) increases, once for x=0 where f(x) levels off at 29, and 9 more times for positive x values as f(x) decreases. There would be 10 unique primes, but 19 values of x that produce a prime f(x). 
[QUOTE=R.D. Silverman;140306]Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general definition of prime (i.e. not restricted to just N) then f(x) will be prime i.o. (although a proof is lacking). If we allow x \in R, then f(x) is indeed prime the required number of times.[/QUOTE]Thanks, yeah I was assuming x is an element of R. f(x)=398x^2; x element of N To satisfy MiniGeek's requirement of total primes: f(x)=143x^2; x element of Z 
I am obviously not communicating well. However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site. I am looking for "interesting" examples as listed on these web sites.
On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n2+1. Has it been proven that only 7 exist? Wikipedia, states that there is only one positive Genocchi prime; has this been proven? 
[QUOTE=Housemouse;140313]However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site.[/QUOTE]It is usually assumed that you know how to [url=http://justfuckinggoogleit.com/]google[/url] things you are interested in.

[QUOTE=Housemouse;140313]On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n2+1.
Has it been proven that only 7 exist?[/QUOTE] No. Given that (10^n + 666) * 10^(n2) + 1 is not divisible by 2, 3, or 5, a quick guess at the 'chance' it's prime as 15/4 * (1/log(10^(2n2))) The sum of this from 2 to 3000 is 6.98, so having 7 from n = 2 to 3000 is pretty much what you'd expect. The expected number up to a million is 11.72, so it would be unusual if only 7 existed. In fact, since the harmonic series diverges, you'd naively expect an infinite number of such primes. [QUOTE=Housemouse;140313]Wikipedia, states that there is only one positive Genocchi prime; has this been proven?[/QUOTE] MathWorld has "D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.". 
To the original question: Of course there are uncountably many sets of primes (beth_1), a countable number of which sets are finite; but the question seems to be about intuitively 'interesting' sets of primes.
Toward that end I suggest my small compilation here: [url]http://en.wikipedia.org/wiki/User:CRGreathouse/Tables_of_special_primes[/url] 
Additional "interesting examples"
I found additional "interesting examples" at primes.utm.edu, under Rudolf Ondrejka's top ten.
Sometimes finding useful information using google is like looking for a needle in a hay stack. 
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