Handling unsolicited proofs of famous mathematical problems I have been receiving mails from (most probably amateurs), who claims to have proved famous mathematical problems, like the ABC Conjecture or Goldbach Conjecture. But invariably, they all contained mistakes. I decided not to waste my time on such unsolicited documents. But recently something interesting happened.
About 14 days earlier, I have received a mail from an Indian undergraduate student who claimed to have proved the Sylvester-Gallai Theorem in an elementary way. What is more amusing is that he claimed to have proved it using Mathematical Induction and a basic Euclidean Axiom. I decided to ignore it as usual. But yesterday I got his mail, telling me that-

I suppose you haven't considered my document worthy of your time and so you haven't gone through it at all, or it may be that you are so busy that you haven't found time to check your email account. If that's the case then just ignore this mail. But if it's the first case then I would like to tell you something.
Perhaps you have heard about the Indian Mathematician Srinivasa Ramanujan. He also sent his mathematical works to renowned mathematicians like Baker and Hobson but they didn't reply. Later he sent his manuscript to Hardy and his genius was recognized. But just suppose that Hardy also considered his work to be the work of a crank, without even going through it. Consider this be the case even if he would sent it to other mathematicians. How long could he continue sending his unsolicited formulas and theorems (which were without proof!) to other mathematicians and be rejected? Of course, finitely many times. After that, he perhaps wouldn't write to any mathematician even if he had, suppose for example proved the Riemann Hypothesis. Why would he? He was likely to be rejected.
So I suggest you at least to go through my document thoroughly and tell me precisely about it.
Please don't behave like Baker or Hobson.

What should I do now? Should I remain silent or go through the document? Any suggestion will be welcomed.
Unfortunately, I think there's little or nothing you can realistically do for most amateurs sending unsolicited manuscripts.  What they don't seem to realize is how common this is and what a bad state most of the manuscripts are in:


I average several amateur e-mails per week (and I shudder to think of how many Andrew Wiles or Terry Tao must get).  If I carefully read each paper and sent comments, that alone would occupy a substantial fraction of my professional activities, so I have to prioritize.
I at least flip through the papers, and most of them are obviously crackpot work.  Occasionally I see one that doesn't look ridiculous, and I try to be encouraging when appropriate, but I have yet to receive a publishable paper from an amateur.  The best I can do is generally to offer encouraging advice, and even that's uncommon.
Some people seem beyond hope (for example, the ones who send word salad), but some could presumably become solid researchers given the right education and mentoring.  However, this is not something I have a lot of time to provide.  I've got plenty of in-person students, some of whom would probably like more interaction, and I wouldn't feel comfortable telling them "Sorry, I'm busy trying to explain to some guy on the internet why his fuzzy understanding of quantum mechanics doesn't actually yield a short proof of Fermat's Last Theorem."  Even if the amateur seems promising, they aren't likely to be dramatically more promising than my students, and mentoring over the internet is less effective, so it's still an awkward trade-off.
Some amateurs react very poorly to feedback.  If you suggest their results are known (while complimenting them on their rediscovery), they angrily suggest that you must not have understood what they meant or are trying to deny them credit for their work.  If you don't believe their results, they accuse you of incompetence or laziness.  If you encourage them to apply to graduate school, they scoff at what academia would have to teach them.  This is of course only a minority of amateurs, but it's just common enough to discourage giving honest feedback: there's too much of a risk of feeling like you wasted time offering feedback to someone who only wanted validation and responded with insults.
Part of the problem is grandiose visions.  When people spend too much time daydreaming about being the next Ramanujan or finding the proof that didn't fit in Fermat's margin, it's really unsatisfying to learn that their story isn't actually as remarkable as they hoped.  It's much easier psychologically to move to the parallel story of the genius oppressed by academia, rather than starting an academic career from scratch.  (And even people who show no sign of grandiosity in their original e-mail sometimes have it hiding below the surface: I imagine that anyone who sends unsolicited accounts of their discoveries to experts is hoping for some degree of acclaim.)


So what to do about this?  In an ideal world, I'd give lots of time and attention to everyone who wrote, but these are scarce resources.  In practice, I handle it this way:


If the paper genuinely engages with my work and shows no signs of craziness (e.g., drawing religious conclusions from mathematics), I give at least a brief reply.  Same thing if I have some other good reason to believe it was sent specifically to me, and not just as one of many recipients.
If the paper looks relatively promising but has nothing specific to do with me, I'll reply if I have time and feel that the reply would be well received.
If the paper is on a topic I particularly know and care about but doesn't involve my work and doesn't seem especially promising, I might reply.
Otherwise, I probably won't reply, and almost certainly not if the paper deals with famous unsolved problems.