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Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for verification or send to some authorities? Is there a high risk that nobody will treat it seriously?

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Not to discourage anyone or anything, but it is almost certain that an untrained amateur's "proof" is erroneous and trivially erroneous at that. Before sending it for publication, they should show it to someone with mathematical training. Errors usually show up as circular reasoning, the use of invalid / undefined mathematical operations, category errors, not to mention the subtle logical errors that can happen to the best of mathematicians. – user_of_math Aug 15 '14 at 7:52
@user_of_math The question says nothing about "untrained". I guess many former IMO medalists, for example, are not professional mathematicians, so they are amateur, but I wouldn't say it is almost certain that their proofs are trivially erroneous. – JiK Aug 15 '14 at 10:46
If you really believe you have something, and fear having credit stolen should you show it to a trained, "professional" mathematician: write down your proofs, seal them, then mail them through traditional mail to yourself (if in the US, use USPS and mail it in a tamper-proof envelope). (You can also get the documents notarized before mailing.) After receiving the letter, do not open it. Take it to a bank, open a safety deposit box, and place the sealed unopened mail in the box. This provides a "poor man's copyright" which is better than nothing. – Doc Aug 15 '14 at 14:19
I hate to see academics poo-poo each other. Too much negativity in these replies!!! So, best of luck to the OP, I hope your solution makes it! – Jim Beam Aug 15 '14 at 15:57
"Mail it to yourself" is advice that's often repeated, but it's likely poor advice. It does not provide any significant level of evidence of the date of authorship since seals and postmarks are both trivial to fake. If you're really concerned about this, consulting a lawyer and using a proper legal service designed for this purpose, or if not that, at least making proper use of a cryptographic notary service that could provide evidence of the date of signature, would be a much better course of action. – R.. Aug 15 '14 at 16:21

I'm no expert on this subject, but the impression that I have comes down to this: your work may be very worthwhile, but it's hard to be taken seriously if you cannot communicate your discoveries.

A particular difficulty in getting published in mathematics is that the subject is very technical -- to use a shaky metaphor, mathematicians speak their own language; and, within the field, each subject has its own dialect. If you're working without formal training or regular interaction with other mathematicians, it's naturally going to be difficult to gain the knowledge of terminology, notation, conventions, etc., that is needed to write research-level mathematics.

The problem is that lack of mastery of "the language" is often construed as a lack of understanding of concepts themselves. A professor of mine once mentioned that he receives many papers from amateurs, and in these there are often give-aways (like misuse of terminology and unintelligible symbolic manipulations) which indicate that reading the paper will simply be a waste of time.

With this in mind, put effort into writing your results clearly and carefully. Strive to use the vernacular of the field in which you have made a discovery. Read/skim papers on subjects close to yours to get an idea how to do this. Post on MSE with terminology questions. If you have trustworthy friends with experience in mathematics, try to get them to read over your proof and give you their candid opinions (both about your exposition and about the validity of your arguments).

Probably far more helpful than my rambling, here is a webpage with extensive advice for amateurs, including suggestions for how to go about publishing. You might also find this MSE thread helpful.

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+1, clear communication is the most important determinant of whether or not an outsider will be heard by insiders. – whacka Aug 15 '14 at 8:25
There is also some excellent advice from Terence Tao on this. – Andrew Kelley Jul 3 '15 at 18:45

I think that there are three things you need to do.

  1. Type it in $\LaTeX$. No matter how good your results, or how clear your exposition, if it is written in Word using Equation Editor then noone will take it seriously and it will be lost to the world. Get the source-code of some good papers off of the arXiv to see how to do this well (find a paper, then go "Other formats $\rightarrow$ Download source").

  2. Establish precedent. I have never quite understood this one thought, but perhaps that is because I am a tad naive. However, I do not think that the best way to do this is to post it on the arXiv, because doing this lets the whole mathematical community see your work and if you have a simple error then you will have lost all credibility (note: you cannot delete papers from the arXiv).

  3. Send it to someone to proofread. However, this is difficult for all involved. Chances are, no professor will have the time to look at your work (they are busy people, who barely have time for their own research!). Thinking outside the box, I think the best thing you could do would be to cherry-pick a few PhD students and send it to them. They will be flattered, and will have more time to respond (or, at least, will not be getting dozens of similar e-mails every week!). If they think the work is worthy, then they will pass it up the tree. Anyway, if a professor does decide to look at your work then they will probably fob off the checking to a PhD student. For example, during my PhD there was a very insistent man who ended up frequenting the pavement outside my department, waiting for mathematicians to leave and then thrusting copies of his work into their slightly scared hands$^{\dagger}$. In the end, one professor took one for the team, and gave his first-year PhD student the challenge of finding the error.

$^{\dagger}$This is not a good way to make friends.

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You can't delegate "taking one for the team". What that professor did was jump under the crazy bus and pull a PhD student under too ;-) – Steve Jessop Aug 15 '14 at 9:05
+1 for this answer. I share your puzzlement about 2. Establishing priority worries "amateurs" a lot although, today, a preprint publicly available on the internet is all the proof one needs. // It seems 2. and 3. are both achieved by sending the paper to a journal for refereeing, no? In every journal whose Editorial Board I am on or whose modus operandi I am aware of, every paper sent to the journal is looked at, whatever the credentials of the authors. True, crackpot submissions usually require a very small amount of time to be rejected, but, ... – Did Aug 15 '14 at 9:22
... despite a common myth, even they are submitted to a proper refereeing process. – Did Aug 15 '14 at 9:23
@SteveJessop To include in the flowchart the task of convincing the author strikes me as quite unreasonable (and this is never a task journals embark on with authors, is it?). – Did Aug 15 '14 at 9:41
@Did: that's why it's (potentially) a crazy bus. Once you've first taken on a "very insistent man", they're not standing outside the department thrusting papers at people, they're thrusting papers at you! You may not think you've made a long-term commitment to persuade them of anything, but short of moving cities it's not necessarily clear what will terminate the correspondence. So in the professor's position and given there's a warning sign in evidence, I think it would be grossly unethical to instruct the PhD student to speak directly to the amateur! – Steve Jessop Aug 15 '14 at 9:46

Hardy to Ramanujan: 'Let me put the matter plainly to you. You have in your possession now 3 [Ramanujan had only two] long letters of mine, in which I speak quite plainly about what you have proved or claim to be able to prove. I have shown your letters to Mr. Littlewood, Dr. Barnes, Mr. Berry, and other mathematicians. Surely it is obvious that, if I were to attempt to make any illegitimate use of your results, nothing would be easier for you than to expose me. You will, I am sure, excuse me stating the case with such bluntness. I should not do so if I were not genuinely anxious to see what can be done to give you a better chance of making the best of your obvious mathematical gifts.' Quoted - p181, The Man Who Knew Infinity, Robert Kanigel.

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Along the lines of this quote (+1), it's worth noting that Ramanuajan sent two failed letters (returned without comment) to English mathematicians before contacting Hardy, who realized his potential. In the book, the author suggests that the reason for these first failures might have been that the mathematicians Ramanujan chose were older and better-established, hence less open to going out on a limb for an unknown. – vociferous_rutabaga Aug 15 '14 at 14:01

There's a list out there that I'm sure someone else will be able to attribute that assigns crackpot points to a paper to determine whether or not it's worth reading ( I think John Baez? ). Explicitly or not, most professionals decide whether or not to read a proof based on the ideas in that list.

At any rate, regardless of what you've proved, and regardless of whether you're professional or amateur, your best bet to get your paper read is write a good summary introduction that introduces the main ideas and insights in your proof. Your first couple pages should lay out the new ideas, the main technical tools used, etc If your insight is obviously flawed, this will help an expert to quickly pick it out and point it out to you. If it's an old idea known not to work ( but maybe this isn't known to you ), an expert will also be able to pick that up quickly and suggest you make sure you have addressed the old attempts. If it's a genuinely new idea, though, then that's what's going to make the expert reader continue reading to the body of your proof.

Agree with frogeyedpeas that arXiv is the best approach. I think going directly to a journal would be very difficult. You can try reaching out directly to some professional academics as well, but I strongly recommend you do not send it to the top people at top Universities. Someone at a local college or University is a good bet. If you've been able to formulate the new idea yourself without professional training, probably anyone at any University working in math will be able to follow it.

Also, maybe goes without saying, but do write it up in LaTeX. No one is going to read a Word document proof.

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You mean this right? – William Aug 15 '14 at 8:10
@William yes, exactly. I forgot it was for physics, but the same principal ideas apply. Largely, be humble in your writing. – Callus Aug 15 '14 at 8:12
Another useful list in this vein is from Scott Aaronsen's blog: Ten Signs a Claimed Mathematical Breakthrough is Wrong. – Semiclassical Aug 15 '14 at 11:14
There's an adaptation of the physics list to mathematics. – apnorton Aug 15 '14 at 21:28

If an amateur has created a proof of a problem the natural step is to publish it on either arXiv (or if you don't have access) viXra.

At that point you have documented you were the first to get the idea at a certain time. Now if you think it's material worthy of appearing in a paper you should attempt to reach out to relevant journals (which will have a standard method of submitting articles, etc.) and then submit your proof.

Additionally, you may want to reach out to relevant professors and researchers in that particular area to have them look over your stuff (this may take some time to coordinate so be patient!).

Again to be taken seriously in both areas is tricky. If the proof is nothing massively groundbreaking it shouldn't be a problem, but if you hypothetically did prove something huge like P=NP then chances are a good 99.9% of people will laugh you off (if you are an unestablished amateur) regardless whether you are correct or not.

This shouldn't be taken negatively, it has nothing to do with the mathematics community, it is human nature.

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I would not recommend vixra. If the OP is concerned with protecting a claim, one can put the paper on one own website. Although vixra philosophy of allowing open publication is commendable, the experts are usually not in the audience of vixra so vixra is not a good place for dissemination. Moreover, there are many very low quality papers on vixra. Usually, the purpose of publishing in a good journal is to earn the distinction of being among reputable works. It is not usually a good idea to put a paper among the terrible ones on vixra. – William Aug 15 '14 at 7:50
The founding idea of vixra is interesting and there may indeed be some good papers on vixra. However, there are numerous papers with serious flaws and error. You may disagree with this point, but this is actually irrelevant. What is important is that vixra has an awful reputation. If the OP is an amateur having difficulty getting recognition, why would you shoot yourself in the foot by putting a paper on a place which a significant number people view unfavorable, especially when there are numerous other venues, like scribd, with a positive or at least neutral reputation. – William Aug 15 '14 at 8:03
@William: the amateur will get recognition if their proof of a "well-known open problem" is confirmed. They're hardly going to be known as "that guy who solved the Collatz conjecture, but then threw it all away by publishing on vixra so I'm not going to bother reading what he has to say about Goldbach" ;-) Journal rep is for when your result is obscure or otherwise difficult to evaluate directly. – Steve Jessop Aug 15 '14 at 8:52
Also, they would still be known as "the guy who solved it and put it on vixra." If I were the type to fantasize about being the guy who solves the Riemann Hypothesis out of nowhere (cough), my major worry would be the loss of style points inherent in neighboring with 75% crazy train. – whacka Aug 15 '14 at 8:58
@Steve while the exact number of five may be an exaggeration, it's certainly no exaggeration to claim that the solution of any famous open problem will almost surely only be understandable by a relatively small community of busy elites. And indeed, I'm pretty sure the name "vixra" will influence most mathematicians' decision of whether or not to seriously read something. They have to make these choices, and they have to base these choices off of something, after all. (One of my friends used to go on vixra literally just for teh lulz.) – whacka Aug 15 '14 at 9:07

If you have solved an open problem and can monetize it then this question is easy. For example, consider factorization.

If you really developed an algorithm for efficiently factoring large numbers, you could write a paper in latex describing your algorithm and proving its correctness. You could send that paper to me. It is likely I will not understand your paper and nothing will happen. You could send it to leading researchers in the field and they will probably think you are a crank and not read your paper and nothing will happen.

Or you could try some of the RSA challenges. Collect a bunch of prizes and likely people will be coming to you asking you questions.

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If you left out the money part, I'd say this was a good suggestion. A lot of open problems are constructive, such as factorization, P=NP, discrete logarithm in polytime, etc. If someone said they had a constructive solution to one of these problems, I wouldn't pay any attention to it without an implementation, regardless of how intractable it might be for large cases. – DanielV Aug 17 '14 at 19:49
@DanielV Exactly. People at my level can evaluate constructive implementations. Otherwise you are restricting yourself to a small elite group who probably don't have time for you. – emory Aug 17 '14 at 20:20

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