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.