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

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.

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.

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.