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I'm puzzled why this question on characterizations of primes was closed as "nonconstructive" or "not a real question". I cast the first vote to reopen and I encourage others to do so.

Please note that the question was posed by a new user (joined 4 days ago). I think one should be a bit more forgiving with questions posed by new users. See Gerry Myerson's comment on Giuga's conjecture for the sort of constructive and real answers that are possible.

Note that the AKS primality test might have been discovered by pondering questions like this. Such questions are surely constructive by my definition. Indeed, Paolo Ribenboim has written many chapters in many books on such topics.

To those who voted to close: why?

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    $\begingroup$ I have also voted to reopen. I don't see anything wrong with us compiling a list of theorems/conjectures of the form, "$n$ is prime if and only if ...." $\endgroup$ – Gerry Myerson Apr 21 '12 at 0:12
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    $\begingroup$ I agree with Bill. $\endgroup$ – Álvaro Lozano-Robledo Apr 21 '12 at 0:34
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    $\begingroup$ I'll vote to reopen, but note that "many chapters in many books [have been written on this]" does not indicate, in and of itself, that it is a good question for this site. The FAQ specifically says "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much." $\endgroup$ – Arturo Magidin Apr 21 '12 at 4:02
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    $\begingroup$ @Arturo Ok, title is now "less accusatory". But meta is for opinions. $\endgroup$ – Bill Dubuque Apr 21 '12 at 4:24
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    $\begingroup$ @Bill: Thanks. ${}{}{}$ $\endgroup$ – Arturo Magidin Apr 21 '12 at 4:25
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The reason I cast a close vote (as "not a real question" {I don't agree with the not constructive reason}) is that it is overly broad: "using algebra, modulus, integrals and other things". Basically, content likely enough for a whole book!

In the mathematical sense, the question is real, but not every real mathematical question is a fit for the Q&A format of this site.

For instance, do you expect people adding answers to check all the previous answers to see if they are duplicating effort? What about the people who are actually voting on the content? Do you want them to read and then vote on the possibly hundreds of answers?

Closing a question does not mean it is gone forever. The user can edit the question (perhaps narrow down the scope etc) and the question can always be reopened.

Also, I don't agree with "be more forgiving" to newer users when it comes to closing questions which we think ought to be closed. We can always educate the user after closing, and once the user shows sufficient interest/motivation to edit/cleanup the question, we can always reopen.

As an aside(not necessarily in regards to this question), people seem to forget that we are first trying to give an answer to OP, and not an answer to our own personal interpretation of the question. The OP might be asking something (which ought to be closed), but many times I have seen people reinterpret it to suit an answer they have in mind and get it opened. This only encourages more closeworthy (for our Q&A format) being posted (the broken window theory). Of course, what questions ought to be closed etc is subjective and meta is the right forum to discuss specific questions.

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    $\begingroup$ Thanks for the explanation. Regarding new users, if one is too unwelcoming then we risk losing potentially valued contributors. I'm not sure what your point is re: "checking all previous answers". What does that have to do with this closure? As for questions reinterpreted by answerers, sure that happens, but I didn't see it here. The question is broad (by design), but it doesn't seem to me to be a question that could be reinterpreted far from the intent of the OP. Why do you think so? $\endgroup$ – Bill Dubuque Apr 21 '12 at 1:24
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    $\begingroup$ I would like to have seen the user given more time and more constructive encouragement to improve the question. $\endgroup$ – Gerry Myerson Apr 21 '12 at 1:43
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    $\begingroup$ @GerryMyerson: Unfortunately, in this case, I don't see how to change the question to make it less broad and getting OP have their question answered. $\endgroup$ – Aryabhata Apr 21 '12 at 2:17
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    $\begingroup$ @BillDubuque: Questions which call for too many possible answers are not suitable for this site. I was just trying to clarify why. The reason for closure is that it is too broad. One of the reasons why we consider broad questions NARQs is the paragraph about voting etc. Only wanted to emphasize that because people tend to think that all mathematical questions are on topic on math.se. $\endgroup$ – Aryabhata Apr 21 '12 at 2:19
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    $\begingroup$ @Aryabhata I don't think the question is too broad (whatever that means). Nor do I think "too broad" should be a reason for closure. Mathematicians frequently ponder "broad" questions. Some of Hilbert's famous problems probably fit your definition of "too broad". Would you reject them too? What may seem broad to one person may seem very specific to another. It's all relative to one's level of knowledge, viewpoint, etc. Valid mathematical questions shouldn't be censored based on arbitrary subjective criteria. $\endgroup$ – Bill Dubuque Apr 21 '12 at 2:23
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    $\begingroup$ @BillDubuque: Exactly what I mean by "subjective". I think it is too broad (constrained to whatever limited knowledge I have), and cast a close vote. You don't and opened this meta thread. The issue is not about rejecting it as a valid mathematical question. The issue is about rejecting is as a valid question for this site, which is geared towards specific questions and specific answers. There is no censoring going on. No one has deleted the question, and it is on the verge of being reopened. Just because my knowledge is limited, do you expect me to not do what I think is good for the site? $\endgroup$ – Aryabhata Apr 21 '12 at 2:34
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    $\begingroup$ @Aryabhata I welcome your explanation. My points above about level of knowledge, viewpoint, etc. were not meant to apply to you or anyone in specific. $\endgroup$ – Bill Dubuque Apr 21 '12 at 2:39

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