# I need to use Lemma X in a paper, but don't have space to include a proof. Is it acceptable to post a proof at math.SE and link to it in the paper?

I need to use a lemma in a paper. Here it is:

Lemma: Any $m \geq 52$ can be written in the form $m=\binom{n+1}{2}+a-3b-6d$ for $n \geq 13$, $a,b \in \{0,1\}$, and $d \in \{0,1,\ldots,2 \lfloor n/2 \rfloor-6\}$.

While I've written a proof, I don't have room in the paper to include the proof due to page limits. Also, the proof is not relevant to the rest of the paper.

Question: Is it acceptable use of math.SE to ask a question What is a proof of Lemma X? and answer it myself, with the intention of citing it in my paper?

It seems better to write "a proof can be found at [math.SE link]" than "we omit a proof" in the paper.

• It might be better to upload a preprint to arXiv containing the paper and the proof of the lemma (assuming the journal you submitted your paper to has no problems with this). – Joel Reyes Noche Jan 3 '16 at 6:09
• I might also consider using a catchall phrase such as Proof. Omitted. (or left as an exercise for the reader). I mean, that binomial is a quadratic in $n$, so it increments in steps of about $2n$. Your $a,b$ take care of anything modulo six, and the range of $d$ seems sufficient to cover the gap between consecutive binomial coefficients. In other words, while I didn't try to prove your Lemma, I will immediately believe it (by the above reasoning). I dare guess your typical reader is capable of similar reasoning. So I would leave this unproved. – Jyrki Lahtonen Jan 3 '16 at 8:01
• Oops. I now notice that there are only four possibilities for the pair $a,b$, so we also need the observation that the binomial coefficient is congruent to one of $0,1,3,4$ modulo six. Anyway, the point stands. – Jyrki Lahtonen Jan 3 '16 at 8:04
• Marginally related: Math SE references in thesis This was somewhat similar situation, although the OP was writing thesis and not an article. (And they had much more parts omitted.) – Martin Sleziak Jan 3 '16 at 9:20
• BTW I wonder whether this would be on-topic on academia.SE. (In which case it might be a more suitable place for this question.) – Martin Sleziak Jan 3 '16 at 10:07
• @JoelReyesNoche I wonder if it's okay to cite the arXiv version of a paper from the published version of that same paper? And as Jyrki Lahtonen points out, it's not particularly difficult to prove, so "proof omitted" is reasonable, and an arXiv paper seems a bit too much for me. (Besides, I don't like having two versions of the same paper.) – Rebecca J. Stones Jan 3 '16 at 10:34
• You make some valid points. – Joel Reyes Noche Jan 3 '16 at 11:44
• One more thing to consider is whether you really want to direct the audience of your paper to a website anyone can contribute to. You may get some irrelevant answers or comments questioning the importance of your question, quality of your answer, etc, all beyond your control. – The Vee Jan 3 '16 at 19:34
• Related to the suggestion of @JoelReyesNoche: See academia.stackexchange.com/questions/49486/… – mweiss Jan 4 '16 at 3:54
• @mweiss, thanks for the link. – Joel Reyes Noche Jan 4 '16 at 4:52
• @RebeccaJ.Stones, it is common to have e.g. a technical report with full details, and a paper with highlights only. – vonbrand Jan 5 '16 at 16:36
• There is one additional and tempting reason to omit the proof. Someone once found a remarkable proof, but he had not enough space in the margin to write it. And this remark made his famous. :-D – Alex Ravsky Jan 8 '16 at 8:06
• @RebeccaJ.Stones: FWIW, I've done both citing an arXiv preprint out of a shortened paper ( emis.de/journals/EJC/ojs/index.php/eljc/article/view/v22i3p40 ) and citing a mathoverflow answer in a thesis ( web.mit.edu/~darij/www/algebra/pbw.pdf ). The thesis was a diploma one, but I guess there isn't much of a difference. Neither of these should be a problem nowadays; and if it is, the referee or advisor will tell you. – darij grinberg Jan 15 '16 at 23:52

Bluntly, I do not care if you intend to cite some post in your paper.

Less bluntly:

• I do not have any principled objection against this practice. You can post whatever appropriate content you please. Your intentions typically are none of my business.

• The content you post should be appropriate, otherwise it could be downvote, closed, deleted. In particular, there should be relevant context and motivation. "I want to cite this" is not. But if you explain how specifically a result is relevant to some research or you manage to motivate the result intrinsically than this is fine.

• Just to clarify for the OP, this answer is (as far as I can tell) exclusively from the point of view of math.SE. The journal you submit to might not be OK with the practice, I guess. So it's an answer for "Is it acceptable to post a proof at math.SE", and we cannot answer "and link to it in the paper". This second question would perhaps be on-topic at Academia. – Najib Idrissi Jan 4 '16 at 10:08
• Yes. Thanks for making this explicit. I purposefully did not comment on whether or not I find it a good idea to write a paper in this way. – quid Jan 4 '16 at 18:55
• I'm going to accept this. I still haven't decided if it's the best possible approach in my situation. – Rebecca J. Stones Jan 5 '16 at 1:02
• As a referee, I would not accept this practice in a paper, for various reasons. But from the point of view of Math.SE there is nothing wrong with posting a question, as long as the question fits the norms of the site. – Carl Mummert Jan 6 '16 at 0:37

In the applied sciences, it is customary to include "supplemental material" accompanying journal articles that contain such things as proofs of lemmas, etc. Is this not the case in mathematics?

If not, I still wouldn't recommend using math.se as a repository for proofs of your lemmas, not only because such questions risk being closed and deleted for being off-topic, but also because math.se is not designed for long-term archival storage of research. I think the suggestion to point to an "extended" version of the paper on the arxiv with additional proofs is a good one.

• Is this not the case in mathematics? In my (admittedly limited) experience, no. I've never seen this happen. There are papers with appendices, but it's still part of the paper and included in page counts. – Najib Idrissi Jan 4 '16 at 10:07
• Additionally to the risks of being closed or deleted, it could happen that someone edit the proof and make a mistake (which is not so good for the credibility of the paper). – Surb Jan 4 '16 at 13:00

I am opposed to this practice. If we allow things like that, would we not also have to allow cranks to post their proofs here?

• A rare downvote for me, so let me explain my thinking. The emphasis should be on what "things like that" we allow, not on some ad hominem characterization of posters. It has happened that I needed a fairly brief but possibly tedious calculation in the course of answering something on StackOverflow or SciComp.SE, and I posted a Question asking for that calculation here, intending to supply one after giving the Community a decent interval to take the "challenge". In many cases I got a better answer than I would have posted. – hardmath Jan 3 '16 at 18:24
• What I assume the OP proposes to do is is to ask the question "How can I prove P?" and then answer it with their proof. If a "crank" posted such a question and then responded with a flawed proof, then the answer would downvoted. Meanwhile, if the question was answerable (ie. the proposition to be proved isn't just nonsense), then someone might post a correct proof. If the proposition was somehow not meaningful, then the question would be closed. I don't see how this flowchart can lead to a problematic outcome. – Jack M Jan 7 '16 at 23:11