I communicate a conjecture and a colleague/professor proves it; are we both authors? Context: I have a (IMO, very interesting, and rather significant) mathematical conjecture. I communicated this conjecture to a professor at my university; he is a respected expert in the topic. He says he has not seen something like my conjecture before, that it is very interesting, and that he would like to share it with his close colleagues. I tell him yes, and that I have some related conjectures.

Now I'm shifting nervously on my couch wondering what will happen.

In the past, even single theorems on this topic (partitions) have been published in journals as high-roller as the Annals. My question: In the event that my conjecture is proven true (by my professor, or one of his colleagues which he sent the result to) and is sent to a journal, even if I did not majorly assist in the proof (I am a first-year undergrad student, I can only do so much!) is it reasonable to request I be listed as an author, since the conjecture is mine and I am the one who shared it?
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.