# I proved a theorem I heard from another researcher. Can I publish my results?

Some time ago I heard a talk in which the speaker mentioned some theorem. I needed this theorem for another paper I wrote. I looked for this theorem in the relevant literature (which I know quite well), but did not find it. I emailed a question to the speaker, he confirmed that the theorem is correct, but did not supply a proof. I also looked at the speaker's working papers, but it was not there. So, I proved the theorem myself.

Now, I think this theorem is interesting and would like to publish it as a stand-alone short paper (maybe in a 'letters' journal). The problem is, I believe the speaker has some proof of this theorem unpublished, so I might be "stealing" his result.

Obviously, the optimal solution is to contact the speaker, but he does not seem to reply. What can I do?

• Your question is unclear to me. the speaker mentioned some theorem To me, a theorem is an already proved statement. What exactly did he say when he mentioned it? A theorem? Or a conjecture? Commented Jun 22, 2016 at 6:01
• Did the Lemma require more space than the margin of Arithmetica?
– Aron
Commented Jun 22, 2016 at 9:03
• He mentioned the theorem and you publish a month later ... bad. After 3 years he has still not published it, then go ahead... Commented Jun 22, 2016 at 13:39
• "Corollary": when giving a talk and mentioning a theorem, indicate the proper way to cite it or do not mention the theorem at all. Exception: very famous or "classic" theorems. Prime example: when the theorem is yours, the audience may not know it, and the date of its acceptance for publication is close in the past or the future. Commented Jun 22, 2016 at 20:29
• "Dear sir, I have proved the theorem you discussed at your lecture. If you are about to publish a proof yourself, I will hold off. If not, or if I do not hear back from you, I will publish." Even if he already published his proof somewhere, you may have proved it in a different way, and your way of proving it may give someone else an important insight. If you find this to be important, chances are others will too. Give all the necessary credit and acknowledgements of course. Commented Jun 23, 2016 at 15:34

If the proof is yours then of course you can publish it. And it is your duty to do it, because the literature is incomplete without it.

Basically you say (but in more formal language):

• at the Holcombe Colloquium (August 2015), R.J. Blenkinsop asserted the following, without giving a proof: insert theorem here.
• no proof was given at Holcombe and there appears to be none in the literature.
• this theorem is interesting and useful, for instance in the context of… insert description of & reference to your paper
• here is a proof

You have thus acknowledged Blenkinsop as the source of the idea and asserted the originality of your own contribution. Both halves of this action are true and ethically sound.

Of course a referee may contact Blenkinsop who, now that fame and glory are involved, may take the trouble to look up his own proof. But unless it's already published ("see my Simple Sums for Simple Minds, page 2"), you still have priority.

• There isn't a "duty" to publish a proof of any theorem, only of sufficiently interesting ones. In fact, I would argue (if only there was a chance anyone would listen to me) that there is a duty not to pollute the literature with junk papers containing uninteresting proofs of uninteresting theorems. So, it all depends on how interesting the theorem is and how nontrivial the proof is. If the proof is not especially difficult and could be found by anyone with expertise in the field, then it's probably not worth a standalone paper. See my answer for additional thoughts. Commented Jun 22, 2016 at 8:07
• How did Andrew Wiles deal with it?
– Aron
Commented Jun 22, 2016 at 9:05
• @DanRomik yes, but aaaahhh Publish or Perish Commented Jun 22, 2016 at 9:31
• @DanRomik The question claims that the theorem is sufficiently interesting. Commented Jun 22, 2016 at 11:44
• Cite your correspondence as well, so that there is no mistake in the fact that you tried to contact the individual to obtain their proof for the theorem, could not find it (from the other works you cited), and so made your own. Commented Jun 22, 2016 at 12:47

Obviously, the optimal solution is to contact the speaker, but he does not seem to reply. What can I do?

You certainly can try to get your proof published as a stand-alone paper, but unless your proof is by itself a major intellectual achievement, I would advise against it.

The problem is that you'll need to acknowledge in your paper that the result has been claimed as a theorem by someone else and you're publishing the proof because you could not locate a proof in the literature. That means you're acknowledging that you're probably not the first person to prove the result, and that will greatly undermine the publishability of your paper. Probably the result is of a kind that any person with sufficient expertise in your area who learns of its existence would be able to prove it. So, despite the fact that quite possibly you'd be making a useful contribution by writing up the proof that someone else hasn't bothered to write, the credit you would get for doing so probably isn't enough to make for a paper you should be proud to put your name to and that would be good for your reputation (it may be publishable in some lousy journal, but I consider that to be much too low of a threshold to aim for).

It's worth noting that there certainly have been many cases (e.g., Fermat's last theorem) where someone claimed a theorem without providing a proof and it turned out later they didn't actually know how to prove it, or the proof was a lot more difficult or interesting than they had let on, and other people had to work very hard to fill the gap. If this is such a case and your proof is something that would be genuinely very interesting by itself even with the knowledge that someone else had already (either erroneously or correctly) claimed the result, then my advice above doesn't apply and your proof could well be worth trying to publish as a stand-alone paper.

Finally, another suggestion is that your proof might be useful to include in a paper you end up writing that includes additional original results that are truly your own. Then the proof doesn't have to carry the weight of the entire paper, and it could serve a useful purpose in making your paper more valuable and potentially increasing its chances to be accepted in a good journal.

• The last paragraph seems like a very good option here, given that the OP even stated that he needed this theorem in connection with another paper (so it seems fitting to include it in that paper). Commented Jun 22, 2016 at 8:40
• Look at it from everyone else's perspective: Someone claimed there is a theorem, and that they have a proof. That's it. That means nobody can use this theorem. It has no value whatsoever. What's worse, you say a proof cannot be published. Whether the proof is hard or not: Everyone else doesn't know this, so trying to prove it yourself might lead to real headache. Publication of the proof, even if it isn't that difficult, solves the problem. Commented Jun 24, 2016 at 14:05
• "Probably the result is of a kind that any person with sufficient expertise in your area who learns of its existence would be able to prove it." This seems uncharitable. Might not 'perhaps' work as well as 'probably', absent any more concrete evidence of the nature of the result or the difficulty of its proof? Commented Jun 24, 2016 at 21:09
• @LSpice I'm sorry you find a completely subjective assessment of probabilities to be uncharitable. By the same logic someone else might find your "perhaps" to be uncharitable, and will suggest that I say instead "perhaps there is a slight possibility that maybe the result might be ...". See my point - where does it end? Since when is "concrete evidence" required to merely suggest that something appears probable to me? Commented Jun 24, 2016 at 22:30
• The question doesn't say that the fellow knew how to prove it. Just that he confirmed it was right. Just like Euler stated regarding Goldbach's conjecture that it was "completely certain theorem" without being able to prove it. Commented Jan 2, 2018 at 11:01

First, I would try discussing this with a couple other people in the area, just to make sure (1) it indeed seems not to be in the literature, and (2) it is worth trying to publish this. (If you want, you can "discuss" with them by sending a preprint and asking for their opinion.)

Then, I would write this up, with part of the introduction being something like "I learned of this theorem from A, but was not able to find a proof in the literature." I can't tell (and I guess you can't either) if A is considering this to be his own theorem or if this is just one of these theorems that experts are aware of but no one has bothered to write up.

Before submitting, I would send this preprint to A to ask if he has any comments, particularly on the attribution of credit to who first discovered the results.

This is a delicate situation, which you should handle as carefully as you can. In particular, I strongly advise against publishing this paper without first discussing it with seniors mentors who know all the details of the situation, and then having a (possibly awkward but necessary) discussion with the speaker if your mentors think publishing could be the right idea.

The problem is that there's a tricky balancing act:

1. If the speaker plans to publish within a reasonable time period, then trying to scoop him by publishing first would be somewhere between incredibly unfriendly and unethical (even if you give him full credit for the theorem statement itself and anything else you learned in the talk). It's not dishonest, but it's still a violation of the usual norms of professional behavior. It would be a major blow to your reputation, and you'd be much better off not trying to publish.

2. If the speaker doesn't plan to publish it, or at least not without unreasonable delay, then it's OK for you to publish a proof while explaining the situation and giving appropriate credit. But in the case of unreasonable delay there's an implied criticism of the speaker, which makes the situation extra contentious. (Don't do this unless you're confident that the community will side with you.)

Unfortunately, nobody can say with any certainty where to draw the line for reasonable delay. I think it's safe to say nobody would consider ten years reasonable, while many mathematicians would consider one year reasonable. (It's common to give talks on work that you plan to write up for publication only after completing a few more things.) A several year delay in writing something up is long but not unheard of. There are also personal factors that can affect what's reasonable: you'll look really bad if the speaker goes around telling people "I intended to write up my proof of this theorem as part of a longer paper, but I got delayed due to my cancer treatments. After 18 months someone who had been in the audience of my talk sent me his own write-up and threatened to publish it himself if I didn't quickly produce a paper of my own."

Maybe none of this matters. It could be that the speaker doesn't care or has no intention of publishing this result, and would be happy to give you his blessing to publish it yourself. It could be that he would be happy to write a joint paper (although you should be very reluctant to propose this, since it too involves awkward issues). The only way to find out what he has in mind is to talk.

the optimal solution is to contact the speaker, but he does not seem to reply

Even if the speaker is reluctant to correspond in general, it seems likely that he would discuss this issue. If you send a reasonable e-mail outlining your perspective and asking what his plans are, and you still hear nothing, then you can try to get in touch by other means. For example, maybe a mutual acquaintance could put you in touch (such as an organizer of the seminar/conference at which he spoke). Of course you need to handle this tactfully, but if you'd like to publish then you shouldn't give up on talking with the speaker if your first attempt fails, since there's too much at stake.

• advise against publishing this paper with first discussing it with seniors .. I have to feeling you were going for without here? Commented Jun 22, 2016 at 16:44
• @Ghanima, in cases like this I encourage you to suggest an edit to correct obvious errors, by clicking the "edit" button underneath the post. Thank you for noting it!
– D.W.
Commented Jun 22, 2016 at 20:14
• @D.W. I am very reluctant with edits that change the intent of the original author (be it perceived or otherwise). I am also working under the assumption that edits <6 characters are not permitted for low rep users. Anyways, thanks for editing it :) Commented Jun 22, 2016 at 20:22
• "After 18 months someone who had been in the audience of my talk sent me his own write-up and threatened to publish it himself if I didn't quickly produce a paper of my own." But in this case, what holds the speaker from quickly replying "I'm working on the paper, it should be out in a year, if you could be patient please."? Commented Jun 23, 2016 at 8:09
• @Turion Assuming the email got through to him, he might be uncertain how to answer or try to answer more properly—not answering at all is bad but common (I'm afraid I'm guilty myself in other scenarios). Commented Jun 23, 2016 at 13:28

Make a collaborative paper with the speaker! Both of you will be stating this theorem and its proof.

UPDATE: The reasons behind this simple proposal are:

• IMHO science and research are a team sport: the primary goal is knowledge, and as an old verse says: "two heads are better than one"
• He's mentioned a theorem in an open/public speaking and his report papers, so basically by the international Bern's convention it is an actual claim, so he definitely has established an authorship right for that theorem if he's the first one speaking about it and formulating it
• But since there's no proof supplied in any report, nor in papers - it leaves the proof claim all yours. Why don't you guys just join your forces and make the world better by opening a proven theorem to the whole world?
• And - as far as I understand - you've proven it while doing your research, so it can be a very fruitful collaboration start for both of you - give it a try!
• This is more of a comment than an answer. Commented Jun 22, 2016 at 14:58
• @jakebeal May be, but in my point of view the full resolution of this situation is that simple Commented Jun 22, 2016 at 15:05
• Can you please explain your reasoning, then? Commented Jun 22, 2016 at 15:33
• Mathematical theorems are not copyrightable, so I fail to see the relevance of the Berne convention. Commented Jun 23, 2016 at 6:30
• I like this answer and the contained advice a lot. But in OP's situation, it seems the other author does not respond properly, so a joint paper seems out of the question. Commented Jun 23, 2016 at 8:01

Here are the main situations where it's ethically ok to write up a result that someone else has already announced. (Of course, even in these scenarios you should be extremely generous in giving the other person credit.)

1. The author has explicitly told you that it's ok for you to write it up.
2. It's a lemma that's not the main interesting part of their work, and you need this lemma as a part of something else interesting that you're working on.
3. The author is being completely irresponsible in terms of announcing results and never writing them up, to the point that it's causing major delays in the development of the field.

The situation you're discussing doesn't sound like it's anywhere close to justifying trying to publish someone else's result.

• From the OP: "I needed this theorem for another paper I wrote." That's your (2). I also feel (3) should apply. The OP's time has certainly been wasted and the other academic's irresponsiveness has been oddly total. Both of these I feel are important factors in a decision like this, more than whether delays in the field are major enough.
– user18072
Commented Jun 23, 2016 at 3:29
• I also think it's not trivial and a bit accusatory to say "publish someone else's result." The OP proved it. Arguably, that's the OP's result, which is a factor in the question. If you disagree with me, I would encourage writing your response directly in your answer.
– user18072
Commented Jun 23, 2016 at 3:31
• OP might have been justified including a proof in his other paper, but if the result is good enough on its own to be a standalone paper, then it's too good for 2. Commented Jun 23, 2016 at 7:10
• But the other author should really answer "I'll soon publish this" and actually do it soon (unlike some of our colleagues ;-) ) to avoid 3. Commented Jun 23, 2016 at 7:31
• @Turion: I totally agree! But not replying to email, and taking a year or two longer to publish than you should are such utterly common venial sins. 3 is really for people who are being totally irresponsible, not just a normal slightly irresponsible. (Though we should all strive to be more responsible.) Commented Jun 23, 2016 at 7:37

In game theory there are `folk theorems`, that is theorems that were discussed but for which no written proof was ever published. Then it turned out there were several ways to prove them. Eventually proofs were published: some buried in books and some as short papers, something like "A New Proof of the Folk Theorem". Could be a way to go in your case.

Depending on the complexity of the proof and the theorem, you need to consider a couple of things:

• Is the theorem original to the speaker?
• Is its validity intuitive?
• Would the speaker have reason not to publish their own proof?
• Does the proof add value to the theorem?

In essence, what it boils down to is that there may be solid reasons you have not been able to find a proof. In fact, this should at least give you enough pause to question the validity of your own proof. Otherwise, if the theorem is only in support of a model, the validity of the model verifies the validity of the theorem without required explicit proof.

• Does the proof add value? Well, if there is no proof, then it isn't a theorem. Commented Jun 24, 2016 at 14:09
• I probably should have said 'knowing the proof'. Some are so trivial that the knowing doesn't truly add value. Commented Jun 25, 2016 at 9:05