I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?
Your question has some issues. Given some of the questions you have asked on other SE sites in the last few days, I have some reservations about whether your question is being asked in good faith, but taken on its own merits it is a reasonable question so I will try to answer it.
The main issue is that, even in asking this relatively simple question, your writing is far from clear. If you cannot write clearly in this situation, your chances of writing up a difficult piece of mathematics or theoretical computer science are less than good. For instance:
His/Her supervisor(s) accept the work and they published it in a highly known journal and they get rejected.
Laying aside issues of subject/verb agreement and consistency of tense, the entire sentence doesn't make sense: you can't publish a paper and get rejected.
It breaks what most people believe
I don't know what it means to "break what most people believe".
or what they have already proven,
What? Are you saying that your proof contradicts other proven results? Taken literally, that would mean that you have shown mathematics to be inconsistent. In practice this could only mean that if your result is correct then some previously published work is incorrect. If that's the case then you need to be very clear about that and explain the flaws in the earlier work. It distresses me that you don't really seem to believe this but are just throwing it off as loose language.
i.e., He/She solves the P vs. NP problem or any other well known open problem.
Solving an open problem would not "break what people have already proven"....that's what it means for the problem to be open. Also saying "P vs. NP problem or any other well known open problem" is a strange bit of coyness: there is no other problem in theoretical computer science (and very few to none in mathematics as a whole) which is "like" P vs. NP. So it doesn't make sense to give that as an example. It's like saying "i.e., he found the Holy Grail or some other famous cup".
In other questions you have spoken specifically about having a proof of P vs. NP and then upon questioning have retreated from this. This sort of vacillation about what you have done is a red flag of "crankiness" that will make professionals wary.
The reviewers strongly reject his/her work with no justification and they said that the result must be wrong.
Saying that the result must be wrong is not just a justification for rejection, it's the best justification. No professional reviewer will say something is wrong lightly. Almost any reviewer who says this will point to at least one specific error. If they do not, then in practice it almost certainly means that the entire document did not make enough sense to them to be more specific.
If your advisor accepts the work, the reviewers reject the work without even explain the mistakes (it is the "best" journal in his/her domain) then what he/she must do?
If you submit a paper to the top journal in your field claiming a solution to the top problem in your field, and your paper does not make sense or does not evince even a correct understanding of the problem, then the editors are likely not to want to spend much time in response. On the other hand, if you are sincerely interested in getting their expertise, it seems reasonable to write back very politely and ask for more specifics about the error. If your response is in any way argumentative then you risk the editorial staff thinking that you will keep hounding them ad infinitum, and at some point they have to stop replying. So you should write back saying that you are not considering resubmitting the paper to that journal but for your own progress it would be extremely helpful to know what is wrong with it. You could also mention that your supervisor found the paper to be correct.
In fact you could be getting more help on this from your supervisor. If you have really "solved P vs. NP problem or any other well known open problem" and your supervisor believes your solution to be correct, why isn't your supervisor moving heaven and earth to be sure your work is getting the attention it deserves? That doesn't add up. The two possible explanations seem to be (i) your supervisor is being too polite with you: s/he does not actually believe that you have solved P vs. NP; and (ii) your advisor's imprimatur does not carry any weight in the community whatsoever. The latter unfortunately means his/her opinion on the correctness of your work is not worth very much.
A good way to find out whether it's (i), (ii) or -- I do admit that anything is possible! perhaps the top journal in your field is unfairly ignoring your revolutionary work -- is to seek your advisor's help in getting another faculty member to evaluate the work, preferably someone in the department that you can speak to recently.
Finally, you seem to have some real worries that if an unknown person solves a famous problem then it somehow doesn't count. This is really not the way academia works, provided the unknown person is capable of presenting the work in a way which makes sense to the experts (and if not, what a shame, but what else could one possibly expect?). Have you heard of the recent example of Yitang Zhang? Zhang was a non-tenure-track lecturer at the University of New Hampshire when he stunned the mathematical world by proving the existence of bounded prime gaps. He submitted his work to the top mathematical journal...and by all accounts they accepted it with unusual speed. In other words, they received a paper from someone they had probably never heard of, looked at it quickly and saw that it was a plausible attack on a huge open problem, and as a result they sprung into action much more rapidly and thoroughly than for most submissions they get. This is an amazing story, but a true one, and it shows how the community responds to a real situation like this.
Regardless of whether the work is correct or not, the following statement applies:
The burden of proof is on the author to convince the reader of the result.
The community (e.g., editors, reviewers) has no responsibility to evaluate your work to your satisfaction. If the reviewers made a good faith effort to read your paper and were not convinced, then you must make your argument more convincing.
(This does not mean, make a few trivial edits and resubmit. This means, prove your results so thoroughly and in such excruciating detail, and with such demonstrably excellent understanding of the problem context, that they become inarguable. Then figure out a way to express the results in a convincing way.)
If in the process of doing so you find an error, well, you'd be in good company.
First, make sure you are not really a crank before trying to convince others. Read these common characteristics of cranks. If they apply to you then get professional help.
For the rest of the answer I will assume that you have really solved a famous open problem. In the following "he" refers to a typical non-expert claiming to have a solution for a famous open problem and "she" refers to an expert in the topic.
There is no easy shortcut for you!
If you are looking for a simple easy shortcut to get your solution verified by an expert then this answer is not for you and I can assure you what you want is not going to happen.Understand the magnitude of your claim!
E.g. If you are claiming to have a proof of P is not equal to NP then you are the guy who is claiming to have a design for a rocket that can be built with the currently technology and resources to take a human to Andromeda and back safely while experts are having hard time sending a human to mars. If you are claiming to have a proof of P is equal to NP then you are the guy who is claiming to have a time travel machine.Understand why experts are reluctant to directly engage non-experts.
Many experts would be interested to know about any major progress in their field. E.g. there are complexity theorists who do read every P vs. NP related paper posted on arXiv (arXiv has a very lenient acceptance policy regarding P vs. NP claims). They will definitely let other experts know if they notice something interesting. ButYou are not the only one with such claims.
There are thousands of people who regularly make such claims.All previous ones suffered from trivial issues no expert would have made.
It is your job to show you are not one of them.Her time is valuable.
For most it is not really monetary. But I think giving some numbers would be helpful. In my university a graduate student is paid over $40/hour to mark simple undergraduate assignments. This is nothing compared to what an expert might charge for consulting in the industry.Non-experts often lack basic skills and knowledge to understand her replies.
E.g. he lacks mathematical maturity, he does not know basic definitions and terminology, etc. It is not uncommon that an expert tells a non-expert what he has is not a proof. She does not mean the proof is incorrect, she means it is not even a proof in the sense that an apple is not a proof. He does not understand when he is told it is "not even wrong!". To make him understand her reply she would have to teach him those required skills and knowledge, too much work just to convince him he does not have a solution. Often he is not patient nor interested in learning (e.g. reading a textbook), he is only interested in a confirmation of what he believes to be a solution. Way too much work in that case.It is often impossible to satisfy him.
Because of the points mentioned above, he often insists on the validity of his claim even after she tells him it is not. At other times where he understands the reply he considers it a simple easy-to-fix error, not a fundamental one. He tries to fix it and get her verify it. This leads to back and forth.He underestimates the required time and effort on her part to answer his claim.
He thinks it is a simple easy job for her to answer his claim. E.g. he expects her to give him a counterexample where his algorithm fails. Finding a counterexample for an algorithm is a very difficult task (as anyone who has marked undergraduate algorithms or complexity theory assignments would know). Finding an explanation why an idea is fundamentally flawed and cannot work is even more difficult.
He does not understand it is not a puzzle.
She is not interested in the question just for its own sake. She expects the solution to the question will be accompanied with major advances in her field. E.g. complexity theorists do not care about P vs. NP just for its own sake. They expect the solution for P vs. NP will come with major progress in our understanding about the nature of efficient computation and its limitations. Often he does not understand this. He thinks of the question as a game or puzzle that he thinks he has won and that is it. This attitude is frustrating for experts.
Now here are some tips:
Be humble.
It is much easier to get her to have a look at your solution if you are genuinely humble and eager to learn and accepting if you are told that you are wrong.Make sure you understand what is required to solve the question.
E.g. understand that a program that seems to efficiently solve an NP-complete problem is not a proof, understand that an idea does not make a proof, make sure you understand the definitions and terminology, etc.Know the basics.
I keep repeating this: read a good textbook on the topic and solve its exercises. It is beneficial for you as you will know more and will be more convincing. It is beneficial for her because you will not waste her time with simple mistakes that you would have noticed yourself if you had read a good textbook. It is annoying to deal with people who claim to have solved P vs. NP but repeatedly make basic mistakes that a good student who has taken an undergraduate course on the topic will not make.Use your real name.
Not using your real name indicates that you are trying to avoid suffering any potential negative consequence of your claim being incorrect. Using your real name indicates that you are sure enough to be ready to suffer potential negative professional consequences if you are mistaken, so you can be taken more seriously. If you are not completely sure about your claim do not waste her time.Don't shirk work. Do your share before expecting help from others.
If you want her to look at your solution you should spend 10 times more time and effort than she will spend helping you. For claims about P vs. NP you have to do way more.You will not get more than one chance.
Make it count. If on the first page of your paper she finds a silly mistake or a basic error (e.g. you do not even know the definitions of P and NP) then she will be done with your claims forever.Understand the known obstacles for solving the question and why they do not apply to your solution.
E.g. if you are claiming P is not equal to NP then you should have a good idea why relaltivization and natural proofs barriers do not apply to your solution. Similarly if you are claiming P is equal to NP.Try to prove simpler more acceptable claims.
E.g. if you have a proof of P is equal to NP then you should also have a proof of simpler weaker major results like Factoring is in P. If you can extract a clean proof for such claims then you can first try publishing them. Such results can be much easier to get verified as they are considered more likely.Make sure your solution is not too strong.
In other words, make sure it does not contradict other known results. E.g. if your argument for P is equal to NP would also show that P is equal to ExpTime (which we know is false) then you are in trouble (Scott Aaronson mentions a few more cases of too strong results in his blog post Eight Signs A Claimed P≠NP Proof Is Wrong).Check your solution.
Make sure there are no mistakes. All steps should easily seen to follow from the previous ones. Make sure you do not make extra assumptions at any point.Recheck your solution.
Put your proof aside completely for two weeks or more. Do not think about it. Then go back and recheck it with a fresh mind as if you were checking someone else's solution.Build evidence for your claims.
E.g. if you have a really efficient algorithm (i.e. its running time is a polynomial with small constants) which you have proven to solve an NP-complete problem then it should not be a difficult task to beat the state-of-art SAT-solvers or to break various cryptographic protocols based on hardness conjectures (those conjectures will be false if P is equal to NP).Write easy-to-read concise clean abstract and introduction.
Do not put any unnecessary background/history/philosophical consequences/discussion of importance/general commentary. It is a famous open problem; every expert knows its significance. Save them for your final version. Right now you should focus on convincing her that your claim is correct. She first wants an easy-to-read short error-free convincing high-level explanation of your solution. It should also explain why any known obstacles do not apply to your solution. It should also contain any other evidence that can support the correctness of your claim. If you fail the reader is not likely to continue reading.Make sure the rest of your paper matches your abstract and introduction.
If you fail the reader is not likely to continue reading.Make sure every detail in your paper is correct.
Follow the standard structure of papers in the topic. Check a few famous well-written papers in the area that have solved major open problems. All definitions should be clear, easy to understand, and rigorous. Every theorem (lemma, etc.) should be clearly and rigorously stated, and the proof of each of them should follow their statement. She should be able to see why each claim in the proof is correct based on the previous steps, definitions, and lemmas without too much trouble. If you fail the reader is not likely to continue reading.Have a general expert who personally knows you check your solution.
I am assuming that you do not know personally any expert in the area of the question. The closer the general expert is to the area of the question the better it will be. E.g. for P vs. NP, you can ask a mathematician, preferably a theoretical computer scientist. Opinion of people who are not experts in the topic may not have much weight but it will make sure you are not making some simple mistake.
Understand that at this point someone who does not know you personally has no reason to check your solution.Have another general expert who knows you personally check your solution.
Rome was not built in a day. You have to build confidence in your solution little by little. Those you convince can become your bridges to reach the experts.If they are convinced ask them to show your solution to an expert they know.
E.g. for P vs. NP, ask them to show it to a complexity theorist they know. At this point you are less likely to be making a basic mistake and you have good evidence to support your claim. Your solution now requires the expertise of an expert in the topic.If she is convinced she will definitely show it to other experts.
News about any major progress in an area will spread very fast among the experts in that area. Other experts (complexity theorists in the case P vs. NP) will recheck your solution independently. If they are convinced you will probably get an invitation to submit your paper to a famous journal (something like JACM in the case of P vs. NP).Do not claim to solve a famous open problem more than once.
As I wrote above, you will not get more than one chance! You do not have a right to ask her to see what is wrong with your fixed solution if you made a mistake. (The exception is when she explicitly asks you to try to fix your solution and send the fixed version back to her.)Do not expect an explanation for why your idea cannot work.
It is unlikely that someone would be able to show formally that an informal idea cannot work. If the idea is formal enough then the reason that it cannot work can be a new interesting result in itself; however, proving such results can be even more difficult than solving the original question. In the case of P vs. NP, if you are claiming to have an efficient algorithm for an NP-hard problem you should not expect her to find an input where your algorithm fails.
In summary,
Understand that she is not required to help you. If she is helping you she is doing so out of generosity. She has a right to stop it whenever she pleases without any explanation. Be mindful of her time, do not waste it for what you could/should have done yourself, try to make her job in helping you as easy as possible, and do not do anything that will make her regret trying to help you.