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This question

Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$

is basically just a statement of a few conjectures by the asker. The "question" is just "What do you think?"

It's not clear what kind of answer can anyone expect for a question like this. (Are we supposed to resolve the conjecture and post the answer on MO? Are we supposed to give hints? When the conjectures are new and due to the asker, then we probably can't point to anything in the literature.)

The conjectures in the linked question are (in my opinion) kind of nice, and deserve a nice home, but is MathOverflow the right place for them? If so, then is "what do you think?" a good way to post conjectures on MO? If not, then what?

This seems separate from the cases discussed in What should be the policy on "open problems" on MO?, which seems more concerned with questions about existing, well-known or not so well-known, open questions, rather than new conjectures formulated by the asker.

The help (https://mathoverflow.net/help/on-topic) has the line "If you want to contribute to (or view) a list of open problems, visit the Open Problem Garden."

So it seems to me that MO questions aimed at introducing and publicizing new conjectures are at least somewhat discouraged and redirected to the Open Problem Garden. But this seems to be unenforced. It could be much more clear.

My concern is that if everyone started posting their conjectures like "here's a conjecture, what do you think?" then... well, it wouldn't be the end of the world, but it would be a pretty noticeable change in MO, and I don't know if people would want that. I for one have several incredibly dull conjectures...

If I'm on the right track, then some possible actions might be to strengthen language in the help doc, and add a flag for off-topic due to conjecture. Maybe something like "MathOverflow is for mathematicians to ask each other questions about their research. See Open Problem Garden to publicize conjectures in mathematics."

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    $\begingroup$ Oh, Open Problem Garden seems to be dying --- last update in 2017, and it won't let me create an account. Shoot. We can't direct sincere askers to a dead site. $\endgroup$
    – Zach Teitler
    Commented Feb 19, 2019 at 7:34
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    $\begingroup$ Basically any question can be rephrased as a "conjecture". I agree that non-questions such as "what do you think?" should be avoided. But banning conjectures altogether makes no sense, we could as well close the website. Really, what's the difference between "Does object X satisfy property P?" and "I conjecture that object X satisfies property P, what do you think?" $\endgroup$
    – Najib Idrissi
    Commented Feb 19, 2019 at 15:29
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    $\begingroup$ @NajibIdrissi, while what you say is technically true, it seems a little disingenuous. (I hope that doesn't sound harsh; I sought, but couldn't find, a gentler word with similar import. I don't mean to impute to you any ill motivation.) I have occasionally sought examples of a phenomenon, been unable to find them, and asked for examples here on MO; but I think that it is meaningfully different from conjecturing that examples do (or don't) exist, in a way that is probably hard to formalise but easy to recognise informally. $\endgroup$
    – LSpice
    Commented Feb 19, 2019 at 17:04
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    $\begingroup$ I've edited the question to try to be less judgmental about possible (imagined) motivations for asking about one's own conjectures. $\endgroup$
    – Zach Teitler
    Commented Feb 20, 2019 at 4:24
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    $\begingroup$ Perhaps a "conjecture" tag could be enforced, so that one can ignore it easily? $\endgroup$
    – user114668
    Commented Feb 26, 2019 at 16:28
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    $\begingroup$ The post now offers a monetary reward, which seems very much inappropriate. $\endgroup$
    – LSpice
    Commented Feb 27, 2019 at 21:36
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    $\begingroup$ @LSpice. The only thing inappropriate about having a monetary reward for the solution of a mathematics problem is that the reward is not enforced through a cryptocurrency smart contract on a cryptocurrency whose coins are distributed as a reward for solving a useful mining problem. $\endgroup$
    – Joseph Van Name
    Commented Mar 1, 2019 at 3:12
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    $\begingroup$ @JosephVanName, I guess it was sarcasm, but my objection was not to the offering of a monetary reward at all, but to the use of MO as a forum for the offer. $\endgroup$
    – LSpice
    Commented Mar 1, 2019 at 12:52
  • $\begingroup$ My answer was deleted - don't know by whom. So I add comment here. I have asked 27 questions on MO and 14 of them received an answer. Therefore 52% of my questions received an answer. $\endgroup$
    – user21230
    Commented Mar 1, 2019 at 13:11
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    $\begingroup$ @MarekMitros, your answer has not been deleted. $\endgroup$
    – LSpice
    Commented Mar 2, 2019 at 3:12
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    $\begingroup$ @LSpice Perhaps that would be a question for a separate discussion. There was a related discussion on old tea (it is from 2011): Is this question acceptable?: 100€ bounty: Do invariant measures maximize the integral? (And you can find some related discussions on Mathematics Meta and Meta Stack Exchange, but I don't remember this being discussed on this meta.) $\endgroup$
    – Martin Sleziak
    Commented Mar 3, 2019 at 23:52
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    $\begingroup$ @ZachTeitler Open Problem Garden seems to come back again. I contacted the admin, Robert Samal. He created an account for me (auto-accounts are deactivated due to spam), and i was able to add new problems. You can ask him the same. $\endgroup$
    – Mario Krenn
    Commented Mar 4, 2019 at 7:17
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    $\begingroup$ @NicoDean Well, now I feel really stupid and lazy for not writing to the admins before. (I was going to do it eventually, I just procrastinated...) $\endgroup$
    – Zach Teitler
    Commented Mar 4, 2019 at 18:08
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    $\begingroup$ @ZachTeitler Maybe let's try helping the admins reinitiating this website, its pretty amazing and if it is slightly active, it could be interesting again for many people. $\endgroup$
    – Mario Krenn
    Commented Mar 4, 2019 at 23:51
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    $\begingroup$ In my opinion, MathOverflow should not be only interested in reference-requests or graduate-level questions. There are some very talented mathematicians (including some Fields Medalists) here. Novel challenging problems (different from well-known open problems) should be allowed. $\endgroup$
    – Zhi-Wei Sun
    Commented Mar 5, 2019 at 1:29

4 Answers 4

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The approach which seems to be throwing every conjecture conceivable on the wall and see 'what sticks' should be discouraged, IMHO.

There is also a difference between conjectures and conjectures. I think that conjectures falling under at least one of the following categories should be allowed. The conjecture

  • feels like they have been studied before, and poster is looking for reference
  • appear naturally in ones research as a minor lemma
  • looks solvable, but is out of scope for the posters expertise
  • is not posted solely to give partial 'credit' to the poster if the conjecture is resolved on MO.

The conjecture referred to in question seem to fail all these above criteria.

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    $\begingroup$ Your fourth bullet point (about credit) seems a bit harsh to the linked poster, who seems to be a consistent MO participant in good faith and not just seeking credit. I wonder if it is really necessary to speculate on his (or her?) motives to give this answer. $\endgroup$
    – LSpice
    Commented Feb 19, 2019 at 17:06
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    $\begingroup$ A bullet point that I might add in its stead, in elaboration of your first paragraph, is that the conjecture should not be part of an easily deformed family of similar conjectures. One can recognise this kind of 'bad' conjecture when a poster's question changes constantly in response to counterexamples. (I reference here not Zhi-Wei Sun, but other posts, of which I'm sure any of us can think of examples.) $\endgroup$
    – LSpice
    Commented Feb 19, 2019 at 17:07
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    $\begingroup$ @LSpice I think maybe the conjectures in this case better belong to the already long list of conjectures on maths.nju.edu.cn/~zwsun $\endgroup$
    – Per Alexandersson
    Commented Feb 20, 2019 at 10:13
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    $\begingroup$ I'd add one more point. It is easy to come up with conjectures based on probabilistic arguments and empirical evidence, but the criterion should be: why are they interesting? For the same reason, I don't really consider problems like Goldbach's conjecture to be interesting. To prove my point, I just made up a conjecture: Every $6k+5$ prime larger than $5$ is the sum of a prime and a positive square. $\endgroup$
    – user21820
    Commented Mar 2, 2019 at 16:11
  • $\begingroup$ "Looks solvable" seems very much in the eye of the beholder, and it may very well be that Z-W Sun had the preamble about three triangular squares in order to suggest that maybe similar techniques apply. The same applies to the first criterion "feels like they have been studied before", although "is looking for a reference" does suggest a possible avenue to improving Z-W Sun's question. $\endgroup$
    – Todd Trimble Mod
    Commented Mar 4, 2019 at 12:40
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    $\begingroup$ @user21820 I don't think it is easy to pose a nice and novel conjecture. Your "conjecture" that each $n>5$ with $n\equiv 5\pmod6$ is the sum of a prime and a positive square is not new. Hardy and Littlewood had a conjecture on $p+x^2$. $\endgroup$
    – Zhi-Wei Sun
    Commented Mar 5, 2019 at 1:15
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I don't think the quantity of conjectures is the problem so much as the rate and the perception generated.

For the cases that motivated this question, I agree some improvement could occur. One improvement would be to ask for references or new pointers to the literature. The next would be to add some references to what is known (by authors distant from the poster, to avoid blatant promotional efforts; in the opposite direction, the poster can reference his own work on his own user page, and the question can state but not link to the user page). If the question is part of a series, appropriate links to the series should be provided. (Appropriate in this context is deferred to another discussion.). Some more improvement would be to add motivation (what other problem gives rise to this question? What overarching study generates these questions in this fashion?).

Much as I am interested in prime gaps, I do not want to see a random prime gap question every day if there is no thought or perspective behind it, or something I can learn. I would consider one every three to seven days, and if it were part of a series, I would prefer to see two or three parts of the series and a link off of MathOverflow to the rest of the series, especially when nothing essential is added by the other questions. (Essential is also deferred, but with each new MathOverflow question should come a slight shift in perspective, or a tie between perspectives, that makes the body of questions a learning experience rather than a series of chapter exercises.)

If the question is known to enough specialists to be open, that should be stated (and a reference provided, if possible), but then the question should be a relevant reference request ("Where can I read about applying Method X to Problem Y?") so that the perception is that the post contributes to the site rather than detracts.

For those who want to comment further on this issue, I recommend stating examples of well posted questions on MathOverflow, as well as (maybe old versions of) questions where some constructive improvements can be suggested. That way people have an idea how to do better.

Gerhard "Demotes Ideas For Doing Worse" Paseman, 2019.02.19.

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    $\begingroup$ I like the (presumably typo-created) adjective "well posted". $\endgroup$
    – LSpice
    Commented Feb 20, 2019 at 0:58
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    $\begingroup$ @Lspice, thank you. Actually, well-posed is important but different from what I intended to convey. I want an attempt made at showing users good examples from the MathOverflow corpus to follow (or examples where it is clear how they can be improved). Thus I used the term "well posted" to apply to certain forum questions. Gerhard "A Letter Makes A Difference" Paseman, 2019.02.20. $\endgroup$
    – Gerhard Paseman
    Commented Feb 20, 2019 at 15:22
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Instead of using MO to publicize conjectures, I think it is now more appropriate to use other platforms that more specifically caters to this need.

I immediately thought of SciLag upon seeing this Meta post.

From SciLag's introductory page:

SciLag is a free web-based platform that provides tools for mathematicians to collaboratively build a dynamic database of mathematical problems at research level.

The platform allows researchers to share, discover, and discuss open problems in their respective fields.

Our hope is to leverage the power of data to the benefits for the mathematical community by transforming Mathematicians’ visions outlined in their research articles into dynamic objects that can evolve through community's efforts.

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In my opinion MathOverflow should allot sub-portal for open questions or conjectures. This sub-portal should be managed by community - the same which manage MO. The role of mathematicion is to ask questions, put hypotheses, publish conjectures. It would be nice to have some database with such questions. It means that there should be possible to add answers or to change status of such questions.

Please count what percentage of questions on MO have no answers. I bet it is half of them. This probably means that no one knows the answer. Should question without the answer within say a year automatically land in "open issues" sub portal ?

Right hemisphere of my brain is probably dominating, because I focus on intuition in my mathematical study. Therefore I am very much interested in conjectures and open questions. These are ideas which come to the mathematician mind but cannot be proved.

The name could be openMathOverflow (similar to meta-) or MathOpenflow.... how about MathOverkill (joke) ?

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    $\begingroup$ Regarding your conjecture that half of the questions have no answers - if you look at the site statistics, currently 74% of questions on MO are answered. (IIRC this means that they have either an answer with a positive score or an accepted answer.) $\endgroup$
    – Martin Sleziak
    Commented Feb 27, 2019 at 11:02
  • $\begingroup$ Thank you. I found out that there are other sites here. Maybe I will move to philosophy or psychology to find explanation why I get down-votes here... $\endgroup$
    – user21230
    Commented Feb 27, 2019 at 17:50
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    $\begingroup$ There's no such thing as a "subportal" on MathOverflow, since to begin with MO is not a portal. $\endgroup$
    – Asaf Karagila Mod
    Commented Feb 27, 2019 at 19:16
  • $\begingroup$ I see. I read on wikipedia what is MathOverflow. I found opinion there, that it is "almost anti-social network". On the other hand Gerhard Paseman is writing in one comment that "MO is large social training exercise". I see contradiction here. Either MO is "social" or "anti-social". You mention small technical obstacle in obtaining website with open problems. We can add one more sub-page on stackexchange. The task is to coin nice name for it. $\endgroup$
    – user21230
    Commented Feb 27, 2019 at 20:28
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    $\begingroup$ It seems unlikely that a question not receiving an answer implies that no-one knows the answer (although I think we do a pretty good job of enforcing the converse). Also, the resolution to the apparent conflict between Wikipedia's and @GerhardPaseman's comments on MathOverflow may perhaps be suggested by the observation that Wikipedia editors are not necessarily frequent MO users, and @‍GerhardPaseman definitely is. $\endgroup$
    – LSpice
    Commented Feb 28, 2019 at 18:46
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    $\begingroup$ Questions without answers are listed in "Unanswered" which is accessible in the left navigation bar, at least on my desktop browser and my tablet (iPad). (I can't figure out how to see "Unanswered" on my phone.) No need to wait a year. ... I'm not one of the downvoters, but I wonder if some of the downvotes are because you are suggesting a feature that already exists, in fact quite prominently? $\endgroup$
    – Zach Teitler
    Commented Mar 2, 2019 at 4:02
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    $\begingroup$ @LSpice There is no actual conflict between Paseman's comment and what one finds on Wikipedia. "Anti-social network" is quoting from an article in the Atlantic, actually a sympathetic portrayal of MO when it was about one year old, and the phrase was meant to suggest a refreshing contrast to "social" networks like FB (Twitter, Reddit, etc.) with their unending discussions, subthreads, arguments, etc. $\endgroup$
    – Todd Trimble Mod
    Commented Mar 3, 2019 at 22:18
  • $\begingroup$ @ZachTeitler OK, we can see unanswered questions. Still there is difference between "unanswered MO question", "open problem", "conjecture". I will not elaborate on the differences. Therefore it would be useful to have dedicated page for "open problems" and "conjectures". Open Problem Garden is maybe good one but it seems there is very little traffic there (note 6 problems in Algebra). On MO there are 37 new questions each day added. It will be much more convenient to work on the same page with open problems. $\endgroup$
    – user21230
    Commented Mar 4, 2019 at 10:34
  • $\begingroup$ If we had such a subportal for impossible questions, then I suggest calling that site MathOverOverflow (MOO) and having that site be cow themed. $\endgroup$
    – Joseph Van Name
    Commented Mar 31, 2019 at 14:51
  • $\begingroup$ @JosephVanName How about MathSurplus ? $\endgroup$
    – user21230
    Commented Mar 31, 2019 at 20:36
  • $\begingroup$ @MarekMitros. The abbreviation MathSurplus will be for MS students and people who have MS degrees in math rather than Ph.D.'s. $\endgroup$
    – Joseph Van Name
    Commented Mar 31, 2019 at 20:55
  • $\begingroup$ @JosephVanName Do you think that people with PhD use different language than the others ? It is kind of funny believe and beyond reality. $\endgroup$
    – user21230
    Commented Apr 1, 2019 at 6:18

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