I communicate a conjecture and a colleague/professor proves it; are we both authors?
Discovering a new and very interesting conjecture is a cool achievement and not an easy thing to do. Congratulations!
About your question, it’s quite possible that your having come up with the conjecture would warrant coauthorship. But it’s possible that it wouldn’t - it’s impossible to say until the conjecture actually gets proved. The issue is that for some mathematical discoveries, the mere statement of the theorem is the more difficult and substantial part of the discovery, in the sense that once the statement is known, finding the proof is not terribly difficult; whereas for other discoveries, it is finding the actual proof that is the difficult part that’s considered more impressive and worthy of recognition.
And sometimes it’s both: for some conjectures, both the people who discovered it and the people who proved it became quite famous for their respective achievements. One such example that comes to mind is the alternating sign matrix conjecture.
So, for example, if the professor and his colleagues spend the next five years working on a proof of your conjecture and come up with an amazingly complicated 100-page proof, I’d say your claim for being a coauthor is weak to nonexistent (although you should of course be credited for discovering the conjecture). But if your conjecture results in a relatively easy proof of a few pages, you can reasonably ask to be made a coauthor of the paper (or more likely than not you won’t need to ask, they’ll just offer you coauthorship as it would likely be seen as an obvious thing to do). A rather similar situation happened recently with a conjecture in linear algebra that was discovered by three physicists. They communicated their discovery to Terry Tao, and this ended with a joint publication by the four authors that was posted less than two weeks later. See this recent article from Quanta magazine.
Dan Romik’s answer excellently addresses your title question. I think it’s also elaborating on a couple of aspects of how the context you describe affects it.
Most importantly: I think the scenario you sound worried about — where the professor and/or colleagues solve the conjecture quickly and write it up, without your involvement — is extremely unlikely. If that happened, it would be quite unethical of the professor. If a conjecture has been publicly announced, then it’s certainly generally considered fair game for anyone to work on. But if an interesting conjecture is communicated to you privately, especially by a student or a junior colleague, then it would be — at least — pretty bad form to work on it further without keeping the person who suggested it involved.
What I would be doing in your professor’s situation — and I think most academics would do something roughly along the lines — is something like the following: First of all, I would try to figure out how hard the conjectures are. Do I know them, or do I easily see that they follow from other results I know? (Presumably not, based on the reply you received.) If not, ask around some colleagues in case they recognise the conjectures, or see a clear relationships between them and known techniques/results; and also think a bit harder about them myself, and perhaps do a bit of literature search. (It sounds like your prof. is at this stage.) Depending on what I can find/figure out at this stage, then:
If someone recognises the conjecture as known, or as an obvious consequence of existing results (where “obvious” means roughly “if you show someone the results and the conjecture side by side, it’s easy to see that the conjecture follows from the results”), or, conversely, if the conjectures are false for similarly known reasons: Then I’d write back to the student to tell them, and congratulate them on (re)discovering an interesting fact, and suggest keeping in touch about research project possibilities in the future (depending on what kind of projects the department’s programme offers). In this case, we’ve all had a fun problem to solve together, but nothing is publishable. This is honestly the most likely scenario — not just for a student suggestion, but for most questions anyone comes up with. That’s just how research is!
If it looks like the conjectures are not obvious but are approachable using techniques within the student’s reach/background: I would suggest that the student works on this as a research project, under my or a colleague’s supervision. (Again, this will depend partly on how “student research projects” fit into the department’s programme/curriculum.) I’d aim to stay fairly hands-off, and giving no more guidance as the student needed. If this goes well, it could well be publishable, with the student as first or sole author.
If it looks like the conjecture is best approached using deeper theoretical tools, beyond what the student can be expected to master in a short time-frame, but is reasonably approachable using those tools, then I might work on it myself or with colleagues, but certainly also keeping the student involved in the discussions (both to introduce them to the techniques involved, and give them the opportunity of contributing if they get up to speed enough on the techniques). This might well result in a paper, probably including the student as an author. (If the conjecture was interesting enough to spark such a project, then it’s most likely enough of a contribution to merit authorship.)
If I and colleagues can’t easily see how to approach the question at all, I’d congratulate the student on finding an interesting and difficult problem. I’d keep it at the back of my mind, and if I later have an idea on how to approach it, I’d proceed as in case (2) or (3). If it’s interesting enough, I might also mention it to colleagues further afield, and would mention that I got it from a student. This is the only case that could reasonably lead to a solution without the student as co-author: if researchers one or two degrees removed from you hear the conjecture, see a solution, and write it up. Hopefully, I would find out (directly or via the grapevine) that they had solved it, in which case I would suggest they at least acknowledge you by name, and possibly invite you as a co-author. In this case, the criteria from Dan Romik’s answer for whether you deserve co-authorship or just acknowledgement would apply.
In all cases: if the conjectures are interesting and novel enough that a solution could be publishable, I would certainly make sure to keep the student in the loop about anything subsequently done with them; and I think most academics would consider it unethical if the professor didn’t do so.
Check out Beal's Conjecture.
The conjecture was formulated by an amateur mathematician named Andrew Beal who wrote to ~50 number theorists & journals. He got some responses affirming the novelty of the conjecture, and the conjecture is now named after him.
If someone proves (or disproves) the conjecture now, though, I doubt he'll be listed as the author. He'll certainly be cited, but if he's not involved in deriving the proof/counterexample, he won't be an author.