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 The problem is, I believe the speaker has some proof of this theorem unpublished, so I might be "stealing" his result.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