So I don't want to share this with anyone I know unless the proof is reviewed by expert mathematicians and I'm fully sure it's correct or at least seems to be.
This sentence seems to be a bit contradictory in itself. I'm assuming you meant
So I don't want to share this with the mathematical community I know unless the proof is reviewed by expert mathematicians and I'm fully sure it's correct or at least seems to be.
First of all, since you are new to the world of academia, it's reasonable that you are worried about having proper authorship. However, I don't think there is anything to worry about here. Just follow these steps
- Contact a professor of your university who is expert in the area pertinent to your proof. An email such as follows will suffice
Dear Prof. [name],
While working on [...] I found an interesting approach for proving the Theorem [...]. As far as I am aware, this is still an open problem, so I would be interested in knowing if my attempted solution actually constitutes a proof (or at least if it's a valid step towards one).
Would you be interested in discussing the details via a Skype/Zoom call? I could also send you a draft of the attempt if you like.
You could also mention that you'd like to make it into a publication if the proof is correct or salvageable. But that would be of course implied. Saying it explicitly could sound a bit arrogant to some professors.
If the professor agrees s/he will likely find room for improvement. Or will discover an irreparable flaw. We don't know. In any case, based on the amount of work done, s/he might suggest to
a. Publish a coauthored paper with you and him/her
b. Publish a solo paper with your name only (in that case acknowledge the professor at the end)
c. Not publish anything at all
If a. or b.: you are in business! If c., you might try to publish anyway, but this is absolutely not recommended, you might regret it later on. Rest assured that, even if the professor contributed a sizable amount, it's still possible that s/he might propose b. There have been several cases where I've seen this happening (albeit for PhD students).
If a., when the peer review arrives, the professor will likely take care of that. If b., you'll be the corresponding author, but you can (should) still ask the professor for help.
This was the by far most likely scenario. Now let's review some less likely ones that might make you worried.
The professor doesn't want to know anything about it. In this case you can look for a different professor. Or contact a post-doc in your department. You might not know them personally, but the likelyhood of not finding anyone willing to listen is low. If you don't find anyone, I don't really have any smart solution.
The professor "steals" your work. That is, s/he publises the proof without giving you credit for it. I find this extremely unlikely, almost unprecedented. There is nothing to gain for a professor to publish one more paper after they have tenure and there is everything to lose if they are found out about it. If it really happens (I highly doubt it will) you have all the e-mail trail and you can prove that you initiated this dialog with the professor. Usually in every university there is a professor who is sort of a reference point for students (when they have complaints etc..). Go to him/her and explain the issue. I'm sure justice will be made.
You're afraid you will look like a fool for attempting to present an incorrect proof to a professor. Don't worry, this won't happen as long as you don't make any assumptions prior to presenting the attempted proof. It's perfectly reasonable to make mistakes. Worst case scenario, your attempt would be equivalent to the question "why is this not a correct proof? Where is the mistake?". This is a valid question and you absolutely have the right to ask it as a student.
Also let me comment that your situation is not unique. I have witnessed other cases of master students publishing before their thesis. I do not know the details but they had a professor who helped and they got credit for it.
Let me also give a personal viewpoint. In academia we need to work under the assumption that we are all on the same team. Our goal is to get mathematics done, not to get to be the the one who says "first!". The latter attitude would lead to people keeping secrets and severely impair collaborations. Again, it's pretty reasonable for you to worry about this, but try to learn to free yourself from these concerns as you mature as a researcher.