# 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.

• I have no idea how many such emails a mathematics professor receive every day, but I think you can give some of it as exercises to some undergraduate students who have an interest for research to find out the mistakes, it may be fun and a learning opportunity for them. May 12 '14 at 7:54
• I would suggest this gentleman to read up on opportunity costs and expected value. May 12 '14 at 9:21
• I am reminded of the famous quote of Carl Sagan: "But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton, they laughed at the Wright Brothers. But they also laughed at Bozo the Clown." May 12 '14 at 13:50
• @TheMathemagician: How does the correspondent's demonstrated above-average knowledge of a historical anecdote make it more likely that his mathematical manuscript is worth reading? May 12 '14 at 14:19
• Your description seems to hint -- but not say explicitly -- that the proof you got is without value. The Sylvester-Gallai Theorem didn't ring a bell with me, so I looked it up on wikipedia and edited in the wikipedia link. That article gives two different proofs, each taking up about half a page. The first one (from a 1986 paper of Kelly) is completely elementary and uses a little Euclidean geometry. Two pages thus sounds long for an elementary proof, but not necessarily prohibitively long. Is it supposed to be clear that the student's work is of no value? Do you feel this way? May 12 '14 at 16:55

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:

1. 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.

2. 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.

3. 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.

4. 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.

5. 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:

1. 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.

2. 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.

3. 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.

4. Otherwise, I probably won't reply, and almost certainly not if the paper deals with famous unsolved problems.

Continue to treat it as spam, and ignore it.

For every Ramanujan, there are many many thousands of time-wasters.

The reward:cost ratio, weighted by the ratio of misunderstood geniuses to time-wasters, is very very low.

If someone has any ability, they should be able to demonstrate it quickly. And if they've got any sense, they'll realise they need to demonstrate it up front to get taken seriously.

So if someone hasn't put a pre-print up somewhere (much easier to do now than in Ramanujan's day), and has no pre-published material, ignoring them is now an even safer bet than it ever was before.

In this particular case, your correspondant may have already tried posting on Math Overflow, although it may be someone else with the same name. Either way, if you're feeling generous with your time, you could prepare a canned response which went to all such neglected geniuses / timewasters that pointed them to Math Overflow, as a good place to engage with the Maths Research community and demonstrate that they are actually able.

• I don't think he is the same person of whom I have been talking about. This person is an undergraduate. But it seems that the person you have pointed out is a professional mathematician. However, the comment- "..The reward:cost ratio, weighted by the ratio of misunderstood geniuses to time wasters, is very very low." is something useful to me. But do you actually have a statistics or it is just your intuition? May 12 '14 at 8:02
• Just to play devil's advocate: how would you say one should demonstrate their ability up front, other than by sending a manuscript? (Perhaps by submitting it to a journal?) May 12 '14 at 18:34
• @DavidZ: I think the question in this case is: why isn't the student showing the manuscript to faculty at his university? Even if no one there is qualified to read it (which seems unlikely since the topic is combinatorial plane geometry), they can still work with him to find someone who can. May 12 '14 at 18:36

The fact that he compares himself to Ramanujan just gives you all the more reason to ignore his mails.

If his work had any merit, his follow-up email would have been focused on that merit and how it might have been hard to see at first glance.

• I don't think that if someone compares himself to Ramanujan, his works become useless. In reviewing what only matters is the work itself. It is immaterial whether he compares himself to Ramanujan or Gauss. May 12 '14 at 12:36
• @AlfredGauss Sure, but the fact that his choice of follow-up is that comparison rather than an explanation of the merit of his work says a lot. May 12 '14 at 12:58
• but just as the @AlfredGauss worrying about wasting time for a crank, the sender is worrying about being seen as a crank. It's the worry that makes him has to focus on where he is a crank or not, not the content of the paper. And even if he want to say so, how should he replies if there is no question for him? Mar 17 '18 at 7:09

Many years ago, my university used to send letters that explained they received so many proof they didn't have the time to check them all, so each sender received a copy of the previous proof the university had received and asked them to check that one to help the university with their workload. That worked very well.

I think it was my professor in analysis, who received one letter where someone had worked out an excellent approximation of pi as a fraction of rational numbers (I think it was the next approximation better than 355/113). And he found that the result this man found was actually absolutely correct, not quite as mind-blowing as the sender probably hoped, but nevertheless correct, and he replied with a long letter acknowledging the correct results and a list of sources that would help an interested amateur.

That man was a single and outstanding exception. And the OP starter complaining about mistakes: Most of the time things are so bad, there are not even things that could be called "mistakes".

• What's special about approximations of pi as fractions of rational numbers? I just found a really good one: 314159265359/100000000000 May 13 '14 at 12:12
• To be precise, the convergents $a_n/b_n$ of the continued fraction expansion of a given real number $x$ are provably the best rational approximations with denominator $\leq b_n$.
– E.P.
May 13 '14 at 17:08
• @daviewales: That's actually a rubbish and rather thoughtless approximation, since better approximations with much smaller numbers are known. May 14 '14 at 7:42
• OK. I think I misunderstood your meaning. When you said "better than 355/113", I thought you meant "closer to pi", rather than "closer to pi, with small integers and mathematical rigour". May 14 '14 at 9:24
• The idea in the second paragraph is excellent! May 16 '14 at 13:22

Have you considered offering your professional services for a nominal fee? I would think \$250-500 that a starting price for detailed analysis, and potential support of a mathematical proof would be a fair price. Of course for one that will require considerably more effort, that fee could be increased.

If you are loathe to take the money you could always either donate the fees, or return them to the author. The primary purpose of the fee is to filter out the random amatuer submissions that have not been well thought or checked out. I assume that you would not mind doing a few serious reviews a year, if you could avoid the spam.

• Noah Snyder did this for a bit and made some good money; he tells the story here cstheory.stackexchange.com/questions/4489/… . May 14 '14 at 16:13
• Ramanujan was very poor remember? May 16 '14 at 13:23
• @PatrickT - If he was serious about the proof then finding the nominal fee or offering some other service in exchange should not be a problem. The purpose of the fee is put up a barrier in front those who are not serious that is easy to overcome with a little effort so that those that are serious can get through.
Jul 16 '14 at 13:21
• There is a market for this: What I learned as a hired consultant for autodidact physicists Mar 14 '19 at 15:18
• Discuss this with your employer first, you might get into legal trouble. Aug 17 '20 at 7:31

In a reply, suggest a journal to send it to. Then if it gets accepted for review, the reviewer may have an easy task in front of them. (Either that or genius will be recognised.) Everyone will be happy either way.

• Can you suggest some journal to which he may send his approximately-2-page 'elementary proof' of the Sylvester-Gallai Theorem? May 12 '14 at 12:44
• @AlfredGauss Hahaha, well, the burden of journal-finding lies with him. He's got access to the Internet. Let me just point out that he may have no idea how peer-review and publishing work in mathematics. Luckily, nowadays you don't have to explain it to them. It suffices to suggest Google search terms. May 12 '14 at 12:54
• @AlfredGauss: Possibly this one? May 12 '14 at 15:50
• @NateEldredge The source or the length of a proof can't be a basis for its rejection. The AKS paper is only 9 pages en.wikipedia.org/wiki/AKS_primality_test May 13 '14 at 6:14

I guess I don't see the big moral quandary. You're a talented person who has worked a long time to develop his skills, and you are under absolutely no obligation to give those skills and your time away for free to every Tom, Dick, and Harriet. If you want to that's fine, but the fact that you feel pressured to do this is not good.

This person's case is really not that compelling to me: he has proved a result which has already been proved and by elementary methods also (according to a previous poster). Maybe it could be published but for this person to suggest his ability is comparable to Ramanujan's in some way based on this result seems absolutely ludicrous. To me, his appeal to Ramanujan, which is based solely on their circumstances and nationality, seems manipulative, and his comparison of himself to Ramanujan, shows a sort of hubris I find appalling. If Ramanujan had sent a proof of a result which had already been proved by elementary means to Hardy, do you really think Hardy would have given it a second thought? I seriously doubt it. Based on the information given, maybe he has some talent, but I don't see evidence of a world-class genius being lost here.

The reasoning developed in the second e-mail ("please do not ignore the hidden genius") was true when you first started receiving this type of e-mail, which is why you read these first theorems.

However, after a few try, you figured that the genius/spam ratio (as @EnergyNumbers pointed out) was not worth considering all these e-mail (maybe unconsciously...). In short, I think nothing changed with this e-mail.

If you really want to consider all these emails without spending toot much time, as @bingung said, and if you are giving lectures, you can assign them to students. It would be indeed a great exercice to try to demonstrate the theorems are not valid.

A third option, to give you a good conscience, and since the genius/spam ratio is probably really low, you can review 1/10th of the theorems you receive. It will not dramatically decrease the change to discover a math genius...

After getting rid of spam in your email box by the spam filter, pass every unrequested mail through the crackpot index:

It will give points for, for example:

• mentioning Einstein, Feynman or Hawkins. (I suppose mentioning Ramanujan would be the same but in the mathematical field instead of physical field).

• vacuous statements

I don't believe you'll need more then 2-3 minutes for that.

However, also don't forget to take a look at http://en.wikipedia.org/wiki/List_of_amateur_mathematicians, since there is mathematics outside academical mathematics too.

• Yes, obviously I meant Einstien, Feynmann and Hankiws, in the case of physics. They are as equally important as Erdoos, Rieman, or Ramaniuya. May 15 '14 at 12:59

This question has been asked more in the mathematical sciences than anywhere else. An interesting text on the topic (with advice) is A Budget of Trisections by Underwood Dudley. It is probably available somewhere cheaper than on amazon dudley (I found some related work on scribd.com.) If you are dealing with an intelligent and younger person, it might be useful to point out that your time is limited and that they might benefit from reading that text. The lesson I learned there is that practically no older amateur will take your advice when it is pointed out that they tried to prove something extremely hard or know-to-be-unprovable. Today this all happens online, and you should also look at John Baez's crackpot index at http://math.ucr.edu/home/baez/crackpot.html , I assume there must be a mathematical version. (translate Einstein to Ramanujan, etc.)

Ramanujan is actually a difficult case because he was in fact an amateur crackpot, and his contribution to real math is not clear to me. But yes, he was a genius. He did not reveal the methods by which he derived his magic formula, even though I am convinced he could explain it if he wanted to. He did not want to reveal his secret craft, he only wanted the fame. I say dunk it in the wastebasket in this case, and if you ever come across work which sounds scientific tell the author to submit it to arxiv.

• Referring to arXiv doesn't make sense. If the crank has no academic affiliation, arXiv will need an endorser arxiv.org/help/endorsement so the crank will simply return to writing mathematicians to find an endorser. May 14 '14 at 16:09
• You show little knowledge of Ramanujan's story. May 16 '14 at 13:24

I think a good policy would be to redirect those people to Math Overflow. Let them open a new topic to ask what's wrong with their proof.

If it's trivial to find errors then someone one Math Overflow will point out those errors. If their proof actually works I would expect someone at Math Overflow to recognise working proof.

You only need to write a email to redirect people to MathOverflow once and afterwards you can send everyone who sends you unsolicited proofs the same canned email.

• Please, no. We get plenty of these at MO already, and they get closed immediately because their questions are incoherent. May 14 '14 at 16:08
• @DavidSpeyer : If the main reason they get closed is incoherence, the person who asks the question has feedback and could hopefully ask a better question next time May 14 '14 at 16:23
• Please don't send amateurs to MathOverflow. This is explicitly not the purpose of MO, so the question will be quickly closed, leaving the amateur unhappy because they were specifically told to post it there. Even aside from the purpose of MO, it can be difficult to convince cranks that there's something wrong with their proof. One critical professional skill is writing clear, readable proofs that are precise and explicit enough that one can point out unambiguous errors if they are present. If someone doesn't yet have that skill, then trying to sort out their manuscript can be an awful mess. May 14 '14 at 17:10
• @Christian: Maybe a better suggestion would be to redirect those amateurs at first to MSE rather than to MO. If nobody makes a comment regarding the flaw of the proof then he may wish to post his proof in MO. May 15 '14 at 12:14