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

Question

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 The problem is, I believe the speaker has some proof of this theorem unpublished, so I might be "stealing" his result.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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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!

Answer 6

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.

Answer 7

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.

Answer 8

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.