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The tags and have been recently mentioned in this question: There are 1,732 questions tagged both elementary-number-theory and number-theory However, in that question these two tags serve only as an illustration of a more general issue. In this post I'd like to concentrate on the question whether we can agree on the content of these two tags.

I should say that, in my opinion, the distinction between these two tags should not be based on difficulty, but rather on the content of the questions. (Not every question about divisibility is easy. Not every question about analytic number theory is difficult.) Of course, even a question which is about rather simple concept may require advanced tools. In such case tag might be added later. See Retagging after an answer is given.

When I am not sure whether question belongs to some tag, I usually consult the tag-wiki and the tag-excerpt. However, in this case, this does not help much:

For elementary-number-theory we have tag-excerpt:

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

and tag-wiki:

Questions on elementary, or "classic", number theory: congruences, greatest common divisor identities, divisibility, etc.

For more advanced questions, please use the , , , or tags, as appropriate.

For number-theory we have tag-excerpt:

Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

and tag-wiki:

For questions on congruences, linear Diophantine equations, greatest common divisors, etc., please use the tag.

If you compare the situation with elementary-set-theory and set-theory, you will see that the tag-wikis contain a list of topics which belong under the tag in question. This contrasts with rather subjective wording more advanced topics in the tag-info for (number-theory).

  • How do you decide what belongs in and what belongs in ? Is there some relatively simple rule of thumb?
  • Could we perhaps be able to make a list of topics which belong to and list of areas which fall under ?
  • If we are able to reach some agreement about the content of the two tags, it would be useful to improve the tag-info for these two tags.
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    $\begingroup$ I think we should replace elementary-number-theory and number-theory with number-theory and advanced-number-theory (same with the set-theory and other "elementary" ones). In the current state, it is more likely that someone will use number-theory instead of elementary-number-theory. $\endgroup$ – Aryabhata Jan 4 '15 at 22:29
  • $\begingroup$ I've wondered about this for a long time. For a while I would retag number theory questions according to the idea that anything not involving analytic number theory should be elementary-number-theory, but that idea is not cozy with the separate existence of analytic-number-theory. $\endgroup$ – MJD Jan 5 '15 at 5:15
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    $\begingroup$ We're down to $12^3$ and the goal is obviously Ramanujan's number. $\endgroup$ – djechlin Jan 5 '15 at 18:19
  • $\begingroup$ Elementary number theory is descriptive based on the techniques involved (i.e. no advanced algebraic/analytic methods). Inequalities are proven based on combinatorics, basic factorization results, Möbius inversion, and other classical techniques. Erdös' proof of the PNT which uses only elementary techniques is hardly "easy" but it is considered "elementary" because of the tools involved. $\endgroup$ – Adam Hughes Jan 8 '15 at 0:36
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    $\begingroup$ @Adam, if someone posts a question about Erdos-Selberg, I'd say analytic-number-theory would be a better tag for it than elementary-number-theory. $\endgroup$ – Gerry Myerson Jan 8 '15 at 2:59
  • $\begingroup$ @GerryMyerson would you elaborate on your reasoning for that view? (I'm interested) One of the most lauded part of that proof was that it did not use techniques from analysis (though, as was later proven, it does have an analytic formulation). It's name as the "elementary proof" is (to the best of my knowledge) specifically because of the distinction I mentioned in my last comment. In particular, it's my experience that most analytic number theory is called such because of its use of techniques from complex analysis such as residues, function orders, et cetera. $\endgroup$ – Adam Hughes Jan 8 '15 at 3:05
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    $\begingroup$ @Adam, everything you write is correct. But in the context of tagging for this website, I take "elementary" to mean something more like "covered in every introductory number theory textbook", and "analytic" to mean something like "involves estimates, limits, logarithms, and other tools of analysis". I'm of two minds about things like simple estimates for $\sum\sigma(n)$, $\sum\tau(n)$, $\sum\phi(n)$, etc. $\endgroup$ – Gerry Myerson Jan 8 '15 at 3:13
  • $\begingroup$ @GerryMyerson ah, well I suppose for the purposes of MSE, I can see how that could be confusing, especially to students who aren't professionals and don't know the distinction. $\endgroup$ – Adam Hughes Jan 8 '15 at 3:16
  • $\begingroup$ @AdamHughes The elementary proof of PNT means proof without using complex analysis. Somebody more knowledgeable will correct me if this is wrong, but I think I read that trying to avoid complex analysis was customary at the time, but it is considered less important now. Maybe this MO post is also interesting i this context. $\endgroup$ – Martin Sleziak Jan 8 '15 at 7:19
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    $\begingroup$ @AdamHughes In any case, this is not the place for me to expound on the history of NT development. If you feel like writing more about this, there are related questions on main here and here. $\endgroup$ – Martin Sleziak Jan 8 '15 at 8:19
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    $\begingroup$ Since there is no clear support for an "artificial" solution, why not just go with what happens in practice and is self-explanatory, viz. number-theory is for Number Theory? $\endgroup$ – quid Jan 30 '15 at 10:10
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    $\begingroup$ @MartinSleziak this is somebody trying to enforce what does not happen in practice. The idea to have a restriction on an extremly common tag is in my opinion ill-advised. The effort would be much better spent adding more specific number-theoretic subtags. $\endgroup$ – quid Jan 30 '15 at 10:47
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    $\begingroup$ @barto it is in fact very easy to verify that it is not the case, one just has to browse the list. My solution would be to say: The tag number-theory is for questions in Number Theory; since this is a very broad field consider to add a more specific tag too: [list of common subtags]. I will try to elaborate on this later. $\endgroup$ – quid Jan 30 '15 at 12:44
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    $\begingroup$ @quid It seems that discussion about how these tags were (are) used could be quite long. I would suggest to move it to chat so that we do not fill this with too many comments. $\endgroup$ – Martin Sleziak Jan 30 '15 at 13:34
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    $\begingroup$ Sorry, I cannot use chat at the moment, but then I also should not continue this debate right now. I will come back to this later. Only one clarification: I know very well that this distinction does not originate here with you and is encoded in the current tag-wiki. It is still not a good idea in my opinion and should be discontinued/modified. $\endgroup$ – quid Jan 30 '15 at 13:56
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I think the obvious problem is that seems, semantically, like it ought to be a proper subset of . Therefore, users posting elementary questions naturally think they should use both tags. Experienced users who are doing serious business number theory often know how the tags should be used, because they read the tag excerpts carefully, but less experienced users are less likely to do this, and they are the ones posting elementary number theory questions. If it were the other way around, and the advanced users were the ones using the less generic tag (as is the case with and ), then mistakes might occur less often.

To be honest, I think the only way to feasibly enforce the desired usage would be to synonym into something like , while keeping the detour message in the excerpt to direct new users to the other tag. But this may not be worth the inevitable arguments about whether X question is fancy enough to belong to the smart people number theory club. I forsee rollback wars.

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  • $\begingroup$ Regarding your comment that elementary-set-theory should be a subset of number-theory, I'd just like to point out the situation between set-theory and elementary-set-theory, which, as of this moment are disjoint. $\endgroup$ – user642796 Jan 8 '15 at 10:21
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    $\begingroup$ How about we rename the tags number-theory-elementary and number-theory-advanced? It is not ideal, but it will stop the double tagging, and avoid erroneous use of the first tag for questions regarding the technical sense of the term. $\endgroup$ – Andrés E. Caicedo Jan 8 '15 at 18:13
  • $\begingroup$ @AndresCaicedo Maybe it would be reasonable to post the suggestion from your comment as an answer, so that people can vote for it. (It would also be more prominently visible.) $\endgroup$ – Martin Sleziak Jan 13 '15 at 8:48
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I think that:

The tag should not be a subset of , the later should be reserved for more advanced stuff.

My impression is that these two tags have been used in this way historically. (This is also what the current revisions of the tag-wikis say.) I don't think that going through 10k+ questions and retag them manually is a reasonable undertaking. (Although many older post are definitely worth retagging independently of the outcome of this discussion.) And I think that dividing number-theoretic questions in this way might help users who use tags to filter questions which are closer to their area of expertise.

To keep clear what belongs into which tag, we should try to create a list of topics belonging to . If some consensus is reached, we can try to make some reasonable tag-wiki based on it.


Here is my suggestion for topics which could be considered elementary. Feel free to suggest additional topics in the comments. (Or, if you prefer, some parts of the discussion could be moved to chat.) Maybe also some post on the main can help in deciding what topics in number theory are considered elementary, for example, What topics to include on a first course in number theory

Despite the fact that the position where to make a division between elementary and more advanced topics is rather subjective, I suppose that most of us will agree that the first four bullets (divisibility, gcd, primes, congruences) can really be considered elementary.

The tag is intended for basic topics in number theory, typically covered by introductory courses on this subject. For more advanced topics we have , , , or . (These tags are already mentioned in the tag-wiki.)

The topics in this tag include, for example:

  • divisibility
  • greatest common divisor, least common multiple, (extended) Euclidean algorithm
  • prime numbers, prime factorization
  • modular arithmetic, congurences
  • simple Diophantine equations (in particular, linear Diophantine equations)
  • Pythagorean triples
  • Fermat's theorem on sums of two squares, Lagrange's four-square theorem and similar topics
  • quadratic residua, reciprocity laws
  • arithmetic functions, multiplicative functions (Euler's totient function, Möbius function, number of divisors, sum of divisors)
  • continued fractions
  • primitive roots
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    $\begingroup$ Rethinking what else I did in my first number theory course, I could only think of continued fractions to add to this list. $\endgroup$ – Lord_Farin Jan 12 '15 at 20:29
  • $\begingroup$ @Lord_Farin I was although thinking about them. I've edited the post per your suggestion. $\endgroup$ – Martin Sleziak Jan 12 '15 at 20:34
  • $\begingroup$ This can't work because the number-theory tag is already too overloaded. Better to create a new tag for more advanced number theory. $\endgroup$ – Bill Dubuque Jan 12 '15 at 21:36
  • $\begingroup$ Should we include primitive-roots in elementary-number-theory? $\endgroup$ – punctured dusk Jan 24 '15 at 12:59
  • $\begingroup$ @barto I've added them to the list. $\endgroup$ – Martin Sleziak Jan 24 '15 at 13:09
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    $\begingroup$ Perhaps we should make a list of topics for number-theory too. I think of cyclotomic polynomials, (all?) non-linear diophantine equations (e.g. sums of squares and Pell-type equations; there could be overlap with algebraic-number-theory here), asymptotics and estimates of arithmetic functions (if no analysis involved, otherwise tag as analytic-number-theory) $\endgroup$ – punctured dusk Jan 24 '15 at 13:44
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    $\begingroup$ @barto If you wish to do so, feel free to post such a list as a separate answer. (Or edit this post, but I would personally prefer them to be separate. Meaning of upvote/downvotes on an answer becomes unclear, if it contains several unrelated issues.) $\endgroup$ – Martin Sleziak Jan 24 '15 at 13:52
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I will copy here Andres Caicedo's comment. The reason for doing this is that in this way it will be more visible and, more importantly, users can now vote on this so we will see what is the prevailing opinion in the community. (For comments we only have upvotes, for questions we have both upvotes and downvotes.)

I am posting this as a CW-answer - feel free to edit if you think there is need to clarify something or that I somehow misrepresented the comments quote below.


How about we rename the tags and ? It is not ideal, but it will stop the double tagging, and avoid erroneous use of the first tag for questions regarding the technical sense of the term.1

Also Bill Dubuque said something similar in his comment:

Better to create a new tag for more advanced number theory.


1I'll just add the clarification that technical sense here means that the phrase elementary proof is sometimes used for proofs avoiding complex analysis. So the name of the tag might make the impression that it is related to this usage of the word elementary.

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  • $\begingroup$ At the risk of repeating myself; I am against making a distinction by level. To make this as explicit as proposed here is a bad idea, for various reasons, one of them being that I have no interested in discussing what is "advanced" or not. number-theory should exist as a very broad tag, in it there can and should be various subtags, including elementary-number-theory, also algebraic, analytic, additive, combinatorial, multiplicative, probabilistic (or at least some of them) and further tags that exist anyway. $\endgroup$ – quid Jan 24 '15 at 13:53
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I've been commenting a lot, but I suppose there's something affirmative to be said in answer form as well. Traditionally the "elementary" clause has meant "without complex analysis" as affirmed by several sources. Martin pointed out in the comments this wikipedia article and this MO post which give good context to what professionals mean when they say "elementary" in the context of number theory.

I think, similarly to Alexander, the principal issue with the tags is that "number theory" is a catch-all for the entire subject, just as is for things like groups and rings et cetera. But we aren't complaining about topics tagged and , despite this being of essentially the same mathematical quality difference and the fact that there are over three times as many of them and nearly twice as many tagged and . The one tag being the larger categorical tag, the other narrowing down the context. The relationship here is just that and are more etymologically similar, since one contains a copy of the other plus an adjective. In that sense, I think it is reasonable and correct to leave the tags as-is: one gives a larger catch-all, the other has the ability to clarify specifics.

Gerry Myerson has made the excellent point (also in the comments) that here on MSE, the term has come to commonly be used to connote something not literally the same as the professional use of the term, namely that the material involved is more introductory. Though perhaps mathematically regrettable, it makes sense that this would happen given the relative inexperience of those using it. Certainly it's not the greatest sin in the world.

In all cases, the tag serves a purpose in categorizing topics here, and one which is distinct from the wider meaning of . The latter tag has applications to and topics (among others) where the tag would not apply, either in the professional sense or the sense used here on MSE. The tag wikis should certainly be updated to reflect this, but that would be a much more minor point in my estimation.

To sum up In response to the original questions posed I have these responses:

  • Anything that belongs in belongs in it's a subset of a larger category of things, and therefore the only reasonable question about topics tagged with both would be whether or not they truly qualify as .

  • I agree that the tag wikis should be updated to more accurately clarify at least the subject as what counts, and the tag as a more catch-all.

  • I don't think it's a problem that students are tagging with both: they're both appropriate for the problems they're posting.

  • With respect to your view that the distinction should not be on difficulty, I agree: number theory is an entire area, spanning many difficulty levels. It would also artificially introduce the only such distinction among all the tags other than the vs where the line is a lot clearer. Searching the tag database for the phrase "advanced" returns no results and "elementary" is just set theory and number theory (and elementary functions). And unlike set theory, here the adjective "elementary" has meaning, and--in fact--applies both to the material in undergraduate courses and to the wider meaning, so that it is correct to have them both there for number theory, whereas the adjective has no technical meaning at all for set theory.

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  • $\begingroup$ "Elementary number theory" is not "number theory with elementary proofs, only." $\endgroup$ – quid Jan 8 '15 at 12:50
  • $\begingroup$ Leaving this minor, for the present debate, point aside, I essentially agree with what you say, especially, with the content in the "to sum up." $\endgroup$ – quid Jan 8 '15 at 14:25
  • $\begingroup$ @quid no it is not, I meant that as "the subjects studied are not complex analytic questions," that is to say: diophantine equations, congruences, et cetera rather than Perron estimates, funtion orders, et cetera. I wanted to be more brief in the introduction, simply noting some basic things which made each tag have a distinct character so that it would be more obvious that the tags should both remain. Cheers! $\endgroup$ – Adam Hughes Jan 8 '15 at 21:58
  • $\begingroup$ Here is a question (which I do not know enough about number theory to answer): are there general number theory questions that do not fall into algebraic number theory, analytic number theory, or one of the smaller related tags? I ask because I find abstract-algebra is actually used reasonably often for general questions which don't necessarily pertain to group ring or field theory (e.g. "what happens in a structure where + is associative but × isnt?"). $\endgroup$ – Alexander Gruber Jan 9 '15 at 4:59
  • $\begingroup$ @AlexanderGruber Yes. Some examples include questions on Minkowski geometry of numbers--which does not comprise an entire field of its own, nor merit a separate tag; probabilistic number theory--analytic is used almost (and not universally, but predominantly) for complex analytic methods in number theory; Fourier analysis on groups--partially analytic, they are also used in the representation theory on groups with number theoretic applications in a way which would be wrong to characterize as "analytic." (there are others, but I'm nearing the character limit, and you get the idea) $\endgroup$ – Adam Hughes Jan 9 '15 at 5:27
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Martin Sleziak's answer here contains a (non exhaustive) list of topics which fit in . Here is one for .
The reason why I'm doing this is that there should be a clear view on what is not classified as 'elementary'. Many (too many?) questions are being retagged as , which makes me think if there is still something left for or if anything 'non-elementary' fits in or .

  • Cyclotomic polynomials
  • Transcendental numbers and related topics
  • 'More advanced' diophantine equations, e.g.:

    • Pell-type equations

    ('More advanced' needs to be defined more precisely.)

  • The same for solving 'more advanced' congruences
  • Diophantine approximation (Note sure, because this may often go along with continued fractions, which has been classified as elementary.)
  • Asymptotics and estimates of arithmetic functions should be tagged as no matter the level of the analysis involved. Whether an additional tag or should be added remains unclear.

Feel free to edit.

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  • $\begingroup$ The distinction for analytic-number-theory causes way more issues than it solves. One should tag analytic-nt in addition if it is applicable. Also, so, question on arithmetic functions should be tagged either el-nt, or nt, or an-nt, but never more than one of those, depending on what exactly? This is designed to result in bad tagging. $\endgroup$ – quid Jan 24 '15 at 14:12
  • $\begingroup$ It really depends on the question. Perhaps indeed we should tag a question as an-nt as soon as it asks for asymptotics of some arithmetic function. Either way, it will be impossible to avoid questions tagged with two kinds of number theory, but a strict distinction between el-nt and nt seems doable. $\endgroup$ – punctured dusk Jan 24 '15 at 14:18
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The tag should be an all encompassing tag for questions in Number Theory. It thus would naturally contain , , and so on. Granted, this is a broad tag but still it conveys some relevant information. More specific information can always be conveyed by adding more specialized tags, such as the ones just mentioned. The most common of these tags could be mentioned in the tag-wiki(-summary).

The idea to have a 'general' tag for Number Theory that is 'advanced' or 'everything but [some list]' is not a good one in my opinion.

On the one hand, it is not intuitive and creates a need for significant permanent oversight. On the other hand, Number Theory is a quite diverse and broad field, so something just being advanced number theory is still pretty unspecific information and to encourage giving additional information seems desirable regardless.

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    $\begingroup$ I agree, but unfortunately one user has forced the decision the other way by having gone on a massive retagging spree during the past month. I pointed him to this thread and asked hm to wait for a community consensus, but he continued. I also notified a moderator (RobJohn) but received no reply. So much for democracy. $\endgroup$ – Bill Dubuque Jan 30 '15 at 14:37
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    $\begingroup$ @BillDubuque We will see whether that user will respond here, but as he posted some comments in this thread, he is certainly aware of this question. I want just point out that many other users made retags of the same type in the past. Here is a random sample. $\endgroup$ – Martin Sleziak Jan 30 '15 at 14:40
  • $\begingroup$ While the retags are in line with the tag-wiki and in that sense there is no issue, I do agree with @BillDubuque that it is a bit strange to be quite active on this while there is an ongoing debate on the subject. $\endgroup$ – quid Jan 30 '15 at 15:02
  • $\begingroup$ @Martin Well if one user is free to remove them, then another is free to add them back. Maybe I'll start doing that. If you knew this was occuring, then why didn't you say something long ago? It was very wrong that this massive campaign was hidden from this discussion on the matter. $\endgroup$ – Bill Dubuque Jan 30 '15 at 15:05
  • $\begingroup$ @BillDubuque (and quid): I noticed those retags too, but I did not look at them in detail. (I assumed that those are retags of new questions.) Since this thread doesn't give any consensus at the moment, it is difficult to say what would be the best course of action. Maybe a separate post (separate answer), asking users not to retag questions in these two tags until a consensus is reached? Such post would also be much better place to discuss this. So far the comments posted to this answer do not have much in common with the proposal given in quid's post. $\endgroup$ – Martin Sleziak Jan 30 '15 at 15:13
  • $\begingroup$ @Martin When did you first become aware of this massive behind-the-scenes campaign? $\endgroup$ – Bill Dubuque Jan 30 '15 at 15:14
  • $\begingroup$ @BillDubuque I don't recall exactly. As I already said I think that this is not directly linked to quid's proposal. As I do not want to add more off-topic comments, I'd suggest either making a separate post for discussing this or moving the discussion to chat. And I should also say that the best person to give some explanation about actions of a particular user is that user himself. $\endgroup$ – Martin Sleziak Jan 30 '15 at 15:16
  • $\begingroup$ @Martin I am quite disappointed given that you are the OP, and you knew this was going on for some time, but did not update this thread so that the community could become aware of this. It took other users (quid and I) to stumble upon this to realize what was happening - just like what happened with the massive deletion campaign. Secretly forcing decisions like this sets very bad precedents. $\endgroup$ – Bill Dubuque Jan 30 '15 at 15:20
  • $\begingroup$ @MartinSleziak it seems to me that mainly dually tagged questions are retagged, also quite old ones. (While it might not seem like it given the many comments I write, I am a bit short on time ATM so I did not check carefully.) On the one hand, it is not unusual that somebody does this now as the issue came to attention, on the other hand if somebody was seemingly asked to postpone the activity and does not it would not be so nice. (Added in view of latest comment: I do not believe that you have done something wrong here, and it is not that big a deal either IMO.) $\endgroup$ – quid Jan 30 '15 at 15:20
  • $\begingroup$ @BillDubuque: Did you mention this somewhere other than the comment posted an hour ago? If so, I'm sorry; I must have missed it. I will look into this. $\endgroup$ – robjohn Jan 30 '15 at 16:03
  • $\begingroup$ @robjohn Check your email 4 days ago (and check with the mod who deleted my comment to Andres then). $\endgroup$ – Bill Dubuque Jan 30 '15 at 16:03
  • $\begingroup$ @BillDubuque: Ah, that email was lost in a flood of spam emails that I had skipped while under the weather. $\endgroup$ – robjohn Jan 30 '15 at 16:09
  • $\begingroup$ @quid I downvoted this suggestion back in the day. Re-examining my stance (prompted by the comments in Martin's and mixedmath's chat room) I can now confess that I have more sympathy for the cause of stopping users from reaching a placing in NT leaderboard simply by answering tons of questions about linear congruences and/or modular arithmetic. The current tagging policy seems to achieve that somehow (may be yours truly is placed too high), but if we included ENT... So I would sacrifice a little to the God of Gamification. $\endgroup$ – Jyrki Lahtonen Oct 20 '17 at 8:12
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How to differentiate between (elementary-number-theory) and (number-theory)?

Wouldn't things be cleaner if we just ... didn't?

Currently there are only three tags that have elementary in them:

Having a distinction for elementary questions in a topic is more the exception than the rule. Why, for example, don't we have or or ?

I'd argue that it's fairly easy to tell which are which, and that elementary doesn't add information about the question so much as it adds information about the questioner's (often self-assessed) level of knowledge of the subject, which isn't really the purpose of the tagging system.

So ... why not just migrate all tags to ?

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    $\begingroup$ I agree that the idea that number-theory should be advanced should be abandoned. Only, I think we should keep elementary-number-theory though not as a "low level" tag, but as a sub-topic tag. $\endgroup$ – quid Jan 13 '15 at 19:45
  • $\begingroup$ Since you mentioned it as an example, I'll just point out that (elementary-topology) tag was briefly discussed in the past. (But only it chat, the suggestion did not even make it to meta.) $\endgroup$ – Martin Sleziak Jan 30 '15 at 14:20
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    $\begingroup$ Hmmm ... I know downvotes have a different meaning in meta, but I'd be interested why people disagree. It just seems that "elementary-" qualifiers muddle things more than they help. $\endgroup$ – John Jan 30 '15 at 18:56
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    $\begingroup$ Just a small note: Elementary functions are actually a mathematical well-defined concept (see the tag excerpt), so it doesn't apply here. $\endgroup$ – wythagoras Jan 14 '17 at 12:09

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