Some weeks ago I went to a conference with some of the leading minds in my field of research. One of the speakers (professor A) mentioned a problem that bothered him and that was unsolved up to that point. After the talk (when most had already left the room to get coffee) professor B mentioned a possible algorithm to A to solve these kind of problems (myself and some others followed the discussion with interest).
Out of interest in the topic and motivated by the prospect of a novel algorithm, I spent some time working on this question myself - combined the suggestion of professor B with some of my own ideas and indeed found an algorithm that solves the problem. I can even proof convergence, error bounds, etc. The suggestion of professor B was enough for a working algorithm, but I think that convergence and correctness proofs are only possible with the addition of my ideas.
The question is how to proceed now. If I had thought of the initial idea myself it would be obvious that I should write it up and get it published. As it stands I obviously have to get in touch with professor B (and professor A?) though.
Is it too presumptuous to propose a collaborative paper to professor B? Should I also contact professor A? I can assume that he at least also worked on the same problem these past few weeks... I could just mention them in the acknowledgment part of the paper - but that would still feel like I stole their idea.
I could just write them, what my ideas are on the topic - but I would feel more comfortable doing so if they already communicated a will to collaborate with me. As it stands they don't even know I worked on the issue... On the other hand if I write something along the line of "I have found something. If you are willing to collaborate I will share it with you." it feels somewhat like extortion...
Is there some etiquette how to handle this situation or is it just "first come first serve", and I should see that I get the results published? (OK, the latter is definitely rude - but still...)