text:
I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?
I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.
The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem.
The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:

"The proof must be wrong."
"You cannot achieve such a result."
"You do not understand well the notion of ..."

My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes,  what should you do?

In that case, the supervisee is probably wrong in their perception of their own work; detecting that is what peer review is for. But then again, maybe not; maybe the presentation is just poor, or the claim too outrageous for some hearts/minds. Upload to arXiv for the time stamp and keep improving form and submitting. Your name does not (read: should not) matter when submitting an article so being unknown is not (read: should not be) an issue. Being known for half-baked crank stuff, on the other hand, is: avoid creating that impression at all cost!

See also here and here for more advice. And remember that Nobel prizes and centuries of fame went to people no one took serious in their (life) time.

I should add that the "advice" I link is obviously quite opinionated and should be taken with a grain of salt. For all we know, you do have the solution for an important open problem. But you have to take the situation into account and present your attempt accordingly if you want people to take it seriously. The blog posts I link should give you an inkling of how touchy domain experts can be when they have been bombarded with (to them) obviously wrong attempts whose authors do not accept "no" -- for decades. Write for them.

You should read this page:
http://research.microsoft.com/en-us/um/people/cohn/Thoughts/advice.html

Make sure your findings are actually correct by contacting other pioneers in the field (ask your supervisor to do this) 2)publish it in Arxiv 3) wait for the seminar invitations and the world of fame

@Raphael: I'll remind you that the Nobel Prize has only ever been awarded to living persons. I'll also point out that just because they all laughed at Einstein doesn't mean that if they're laughing at you, you're a new Einstein. Good links though -- thanks for those.

"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown." -Carl Sagan (http://www.c2.com/cgi/wiki?TheyLaughedAtEinstein)

@EricLippert: [Nobel:] I'm aware of that; I was thinking of some artists (and fame, not Nobel prizes) when I put the "(life)" there. [Argument inversion:] Obviously. The fact that false negatives do happen is important to keep in mind, if only so a proper amount of common curtesy is extended towards those who try (and seem to fail).

@Raphael "Detecting that is what peer review is for."

@JoelReyesNoche, Of course, with respect to Mr. Sagan, people didn't laugh at Columbus because he thought the Earth was round, but rather because they knew he'd drastically underestimated the Earth's diameter (resulting in an estimated 3,700km journey vs. what would h ave been an actual 19,600km journey). Columbus was only financed because he promised Queen Isabella I and King Ferdinand II trade with Asia, at a time when the pair desperately needed the income from the trade.

@Jigg
"The proof must be wrong."
"You cannot achieve such result."
"You do not understand well the notion of ..."

If you only believe you solved it, you should start by convincing yourself before convincing others. Only when you wrote all proofs and details down with such clarity that there is absolutely no doubt for you, and you KNOW that you proved you should start worrying about convincing other people ;).. And keep in the back of your mind that basically every researcher at some point in his career believed that he proved some Lemma/Proposition/Theorem just to realize later that he made a mistake...

a crank is defined by wikipedia as someone who is unable to perceive they are wrong even when presented with evidence to the contrary. there is no need to do anything if you accept the conclusion of reviewers. if you do not, are you asking for reviewers who can point out the mistake? and can you accept you might be mistaken & accept evidence to contrary? stackexchange has chat forums with some experts & also suggest in your case [you dont mention computer science but its in your se profile] suggest you give [cs.se] a shot.

@Raphael Am I wrong that you cannot upload anything to arxiv if you do not have an university/research center accreditation (if you are working e.g. in the industry) ?

The root problem here is that you've made an unsubstantiated claim, to wit: that you are not a crank.  Yes, there are a few cases (helicobacter, e.g.) where it took a lot of time and effort to change a paradigm, but those cases are famous in part for their rarity.

@ThorstenS.: Sorry, too many negations for me (at this hour).

@PristineKavalostka: I'm aware of such example, and I'm would be quite embarassed would I have participated in the reviewing process for some years. That said, the fact that it does not always work well (because of laziness, external pressure and what not) does not mean it should not, and that's what I claimed. (Arguably, (at least) conference publications are all but worthless [in CS] and "everbody knows this". But of course, nobody wants to openly acknowledge let alone act on it, because everybody involved is a stakeholder.)

One example of a Nobel Prize winning scientist who was ridiculed is Shechtman for his discovery of quasicrystals. [1]He lost his research job and had been told to read a textbook on crystals, similar to what you've been told. It takes perseverance, convincing people 1 by 1 and finding the right journal to publish in. [1] http://www.bloomberg.com/news/2011-10-05/technion-s-shechtman-wins-chemistry-nobel-for-discovery-of-quasicrystals.html

@JoelReyesNoche : They laughed at Columbus... and they were right and Columbus was wrong. (They all knew the Earth is round, Columbus just assumed it's 4 times smaller and so can cross the ocean. He was just lucky there was some land in-between, otherwise he would have starved just as others predicted)

Maybe I'm missing something, but if the problem is just having no feedback, couldn't he just pay someone to review his work? Of course if he pays him, he have to explain to him what's wrong...

see also analogs of P vs NP in history of math MO for some idea/analogies of how it relates to previous very difficult problems in mathematics; there are other issues raised here wrt peer review. see also area51 stem-review proposal; & also this essay Math monster has many refs on the P vs NP problem incl across stackexchange sites/questions & gives some bkg on how/why its so hard.

@Raphael - Next time you may want to consider posting your comment as an answer instead of a comment on the question.

@eykanal: Noted. I posted the comment back on [cs.SE] and did not assume it had general merit (no restriction to CS or P/NP here). By now, there are better answers so I'll leave it at that (?).

If your proof is actually that "P = NP", then your path forward is simple - write a program that actually solves NP problems in polynomial time.  One you have demonstrated that you can break public key encryption, or calculate some unknown Ramsey numbers, people will believe you.

@mbeckish Neither would this necessarily work (even if the algorithm has polynomial asymtotic runtime, it may not be practical at all for common inputs) nor would this necessarily convince anyone (an algorithm may be fast on some instances, that does not mean it runs in polynomial time).

@EricLippert Not strictly true; if they died between nomination and award, they can still be awarded after death

@mbeckish: Raphael is correct; your requirement is likely sufficient, but it's not necessary. Suppose P=NP but the fastest possible SAT solver runs in n-to-the-kajillion time; it still could be impossible to solve all large SAT problems in polynomial time.

@Raphael - I'm sure if it worked well enough on enough inputs, it would encourage people to examine his proof more carefully.

@EricLippert - True.  Just throwing out one sufficient path forward - definitely not the only path, and possibly not even a feasible path.  We have absolutely no details from the OP.

Just post your paper on the relevant stack exchange. It'll get shot down soon enough.

@Rinze, Shechtman had read those textbooks already, he actually had a PhD in the topic. So his example has nothing to do with when people tell those who haven't read a textbook to read one.

"It breaks what most people believe or what they have already proven" sounds quite impressive: What problem did you solve?

Another link to add to the stories already discussed, and to think about: http://www.sciencealert.com/a-purported-new-mathematics-proof-is-impenetrable-now-what

@djechlin: Proving that P = NP would have no practical impact, because proving that there is an algorithm that solves a problem in polynomial time doesn't help you finding an algorithm that solves the problem in the lifetime of the universe.

I kind of wish I could give a bounty to @E.P.'s justification for his bounty. Lovely.

Maybe giving some "hints" on which method you used and which open problem you settled might help the community here to tell either "well, it might be true", or "no, this is a common mistake."
The questions is so dangling in the air, it is nearly impossible not to think that your claim is wrong.

We have alot of examples that reviewers do not grasp the importance of ones work. CNN that is one of the major breakthroughs in Machine learning and Computer vision related tasks, have been rejected in CVPR. or the dark knowledge paper for Jeffry Hinton, one the founding fathers of Deep Learning had also been rejected. go see how influential both of these papers were(and still are). so yeah, thats something to be expected.

Interesting to know how did this ground-breaking research end up?

I think the question is here "should you [generic] accept what the others are telling you that it's wrong - given that they have not provided any actual reasoning and simply mere assertion - on faith, and forgo trying to understand what the objections actually are and why it's wrong?" Or alternatively, "how can you gain something more concrete about what is wrong with your paper, on the assumptions that they read at least some of it and that their rejection is based on spotting something questionable, yet that they have failed to actually communicate what that is back to you?"

Take a step back and wonder if you are living in a  La-la land.

"It breaks what [...] they have already proven". This is technically impossible. Either they didn't prove, or it doesn't break anything.

Why are this - and the closed [duiplicate] in Academia, for instance - not ideal candidates for SE Worldbuilding, if not for SE Physics. Why not for their own WWW?

If the paper is well-written free of mistakes and typos, then don't worry too much.

Relevant question, witnessing the low chance you did solve a big problem: https://academia.stackexchange.com/questions/181336/is-there-a-single-example-of-an-outsider-considered-a-crank-publishing-a-groun?

Your question has some issues.  Given some of the (now-deleted) 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.

Gievn that I myself am a non-native English speaker, I tend to have more tolerence to the writing from a non-English speaker. Y. Zhang may not be a good example in this case. He went to the US in 1985 and received PhD in math in the US. He has lived in the US since then. From what I know, Zhang has no problem in English while the OP may have serious English language problem. I agree with many parts in your answer, though.

@scaaahu: Not being able to speak English well is not a character flaw.  It is however a problem if English is the language that you're writing your papers in.  There is also a distinction to be made between speaking a language imperfectly and expressing yourself poorly.  In my answer I tried not to harp much on issues of grammar and usage.

Some (quite old) discussion -- mostly on the relevance of English language mistakes -- has been moved to chat.

Seperately, there's also the issue that the "top journals" are not always  going to be the ones with the best, most thorough reviewers or editors: if you get more papers, you can just say "no" more quickly with less explanation if you want to. The reviewers might be better-informed (maybe) or better-specialized, but that doesn't mean they're better reviewers than someone who puts in more time to give better feedback and review of a paper. Constructive feedback different than evaluative feedback, as it were. If you're an outsider, constructive feedback is a better value-add.

Hi Pete, you might make a good point that the author's English is far from good. Many referees will also judge the writer's research ability from his English skills. However, I think English proficiency is only slightly correlated with math proficiency.

Isn't the trouble with Learning's use of English also a clue to the next step, which I'm suggesting would be to find collaborators reasonably fluent in both English and Learning's own language as well as cosmology?

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.
But

You 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.

What do you mean by "understand that it is not a puzzle"?

@Jack, here is what I meant: some people treat P vs. NP like a one level computer game with a Yes or No answer that they have to win. We (complexity theorist) care about P vs. NP because we believe that settling the question will be accompanied with significant progress in our understanding of the nature of efficient computation and its limitations, we don't care about it just for its own sake. As Scott wrote once: "[We] like to take P vs. NP as our “flagship example” of a huge class of questions about what is and isn’t feasible for computers, none of which we know how to answer."

"if you a have an algorithm which you have proven to solve an NP-complete problem then it should not be a difficult task to beat..." -- this may be bad advice. There is no reason to think that such an algorithm should be "real" efficient: it may have abysmal performance on all inputs that we can store. Otherwise, good answer. It illustrates perfectly that it takes extraordinary effort to solve an extraordinary problem. Of course, a crank would a) be unable to diagnose themselves and b) refute many of your points because the have these conspiracy theories. (You do assume everbody's friendly.)

+1: What an exceptionally detailed and helpful answer.  I hope the OP appreciates that the generosity of time and spirit that went into this.

@Raphael, you are right in principle (though it happens rarely in practice), I tried to make the statement more precise.

@Kaveh: I think it's conceivable that a witness of P=NP (if one exists) is exactly such an algorithm. It would definitely explain why nobody has found one so far. Thanks for making the change.

Another point on the checklist might be that you should talk about the problem using whatever notation is most standard, give a very standard citation to a statement of the open problem where that notation is used, and refrain from introducing heaps of your own notation for things until they become clearly necessary.

It's not actually true that an asymptotically-better algorithm for a hard problem makes it easy to beat existing software implementations of other algorithms. For example, there's this linear-time algorithm for polygon triangulation, asymptotically better than the O(n log(n)) time straightforward method - but good luck implementing that in any reasonable amount of time! Not to mention the fact that existing implementations often involve a lot of tweaks and useful shortcuts for common / small cases etc. So - I'd retract that suggestion unless OP is an experienced coder.

This is a magnificent answer! Everyone working on the problem should read this. It is like the other, more famous, 'how to tell if you look like a crank' lists on the internet, only much more constructive and helpful. I hope this answer will find a large audience among the target group, although I wouldn't know the best way to achieve that. The promise 'I will assume that you have really solved a famous open problem' which you live up to, will hopefully assure people actually take your advise to heart (especially because of course in most cases they haven't but are not yet ready to face that).

Your point (2) is magnificent.

This answer is outstanding, and it has become my first go-to resource on this topic. Have a bounty! (which I'll award after the week of free advertising is over)

+1 for the subpoints in 3.  However, I don't think the initial advice about reading common characteristics of cranks to self-diagnose is effective.  All cranks I know would argue that it doesn't apply to them (e.g., Dunning-Kruger).

However, if someone has said already, upon receiving the paper, that it's wrong, that suggests they saw at least one actual and concrete [likely significant] error therein. So there is no need to ask them for a 'specialized evaluation' or to 'pay them to fork over hours of time'. If they got it, did review it anyway, and said it's wrong, something tipped them off, and what OP wants to know, I think, is what do you do when that something has not been communicated. The only added 'work' needed on the part of the reviewer would be to communicate more, not to further analyze the paper.

Sometimes something is so incoherent that it is not possible to point to any part that is wrong. It is what we call "it is not even wrong", "it doesn't even parse". Sometimes, based on previous bad experiences with such authors, the expert fears to point out any particular issue in that situation because often such authors treat it as if it is a problem to fix and retry, whereas what she is saying is not "this is the problem with your solution" but "here is just one of the problems with your solution, and there are many more like that".

For math problems, it is possible to have a formal proof and validate it using a proof checker like Coq, an author can spent time and formalize and validate the proof without an expert. Appealing to an export is a short cut for getting acceptance for a less formal proof. An experts needs to judge correctness in short period of time. Some common issues are the author misunderstanding the definitions or have significant gaps in their arguments. In the later one e.g. she can only say "A doesn't follow from B" meaning there is significant gap in the proof (different from saying "A->B is false").

As @Kimball said, almost all cranks (excluding the really cracked ones who just babble nonsense) suffer from Dunning-Kruger, and will never admit they are a crank. Reminds me of an inconsistent extension T of PA that insists that itself is consistent no matter how rigorously you show by T's own assumptions that T is wrong.

FWIW the first link is functionally broken. A quick look on Wikipedia doesn't turn up anything obviously equivalent to replace it, I think it's best to leave it to you.

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.

+1. "Extraordinary claims require extraordinary proof." If it isn't convincing, make it so -- and remember that all it takes is one counterexample to show that you haven't solved the problem, so you really do need to consider every possible edge case before you can make that claim. If there is an exception, you haven't solved it but you may have solved a subset... which may or may not be new information.

@user3439590  If the reviewers understand the contribution but think the English should be improved, they will write something like "This paper makes a useful contribution, but has English writing issues." If the reviewers didn't write that, then you didn't convince them that you made a useful contribution.

@Selfishness_has_equilibrium keep in mind though that It is also  possible that the English barrier might lead you to not understanding some subtle things with some definition and notions. Which could further easily lead to solving a slightly different version of the problem... And especially in graph theory, sometimes a minor detail is the difference between an open problem and an easy exercise in an introductory course....

@Selfishness_has_equilibrium, if poor English skills get in the way of reviewers being able to understand what you're trying to say, of course they will reject it. That goes double for making extraordinary claims (and triple if you claim persecution or compare yourself to Einstein). I would suggest that you collaborate with someone who has excellent English skills to review/sanity check your work and edit it for readability. The cost may be that (depending on how much they contribute) you will have to list them as co-author.

"If in the process of doing so you find an error, well, you'd be in good company." Reminded me of this news report (Youtube)

The problem is, because you don't know what is wrong or why your proof is not convincing, then you're effectively left guessing at random, and random guessing seems to me to run into Contradiction with other principles such as "not wasting everyone's time". Though I suppose if they don't want to tell you, then yes, go guess at random and "waste" their time with 3-4 more thoroughly revised submissions before giving up, because you have to resolve the contradiction somehow and that means you have to burn one principle xor the other and it seems better to burn the lesser than the greater.

@keshlam : '"Extraordinary claims require extraordinary proof."' - agreed.
'If it isn't convincing, make it so' - Agreed. Trouble is, you have to know first, why it is not convincing, and it is impossible to read the mind of someone who literally gives you nothing else to work with than "It's wrong". OP's question, I believe, is entirely about that lack of knowledge component. How do you improve - or even reasonably reject your own folly - your work when you have nothing at all to go on to tell you what is wrong with it?

You get someone to teach you. That starts by admitting you may be completely wrong. If you can't afford to take classes, audit and/or get recommendations for current textbooks and hire tutoring when you get stuck. If it isn't worth that effort, why should anyone listen?

If your interpretation of events is: "I have a heartbreaking work of staggering genius and the only obstacle to acceptance is that I am not well known and the elites are blocking my work", then you're unlikely to get good advice on what to do here or elsewhere.
The problem, as Raphael indicates, is that while it's possible that this interpretation is correct, it's far more likely that in fact your result does NOT solve the major open problem that you think it does.
Once you admit that this possibility exists, then many steps present themselves, all listed in the very good links provided. Reaching out to people who might comment on your work, looking at the literature to see if approaches like yours have been tried before and have failed, seeing if your solution also solves related (simpler) problems, and so on.

yes. few will mention this & maybe it is considered taboo to do so, but as hinted in your answer it seems in some that these grandiose claims could roughly correlate with psychological symptoms/issues eg delusions of grandeur, narcissism, narcissistic personality disorder etc

This can be a feature of bipolar disorder as well.  (This is not to say that a researcher with bipolar disorder can't achieve excellent results.  It's only to point out a red flag suggesting a possible problem.)

What does your advisor say about all this?
If she really believes you have solved this major problem, she should be moving mountains to help you publish and disseminate it.  (It sounds like her name is on it too, so she has an even greater incentive.)  But you've used the rather lukewarm phrasing that she "accepts" it.  Better get her completely on board first, or get her to explain in more detail her reservations (which may indeed turn out to reveal fatal flaws).
The advantage you have over the average crank is that, as a student, you already have ties to the scientific community, through your advisor.   Take advantage of this.  Once you and your advisor are satisfied that your manuscript is of the best possible quality (see ff254's answer), post it on arXiv, and circulate.  Your advisor surely knows experts in the field, and should have enough reputation that she can get them interested in it.
I'm not sure about your field, but in mathematics, this currently tends to be the way that the community handles solutions to major open problems.  You don't just submit it to Annals, have the referees approve it, and then wait a few months until everyone gets their issue of Annals in the mail and is astonished.  Instead, you get the community to study it first.  You convince a few experts that it is plausible enough to be worth their attention, and they look at it.  Either they find a critical flaw right away (the most common case), or they find a lesser flaw that you or someone else fixes, and maybe, gradually, a consensus develops that it is probably right.  That's when you send it to Annals.
One thing that worries me in what you wrote is:

It breaks what most people believe or what they have already proven...

Which is it?   The distinction is crucial.  If it contradicts people's intuition, that raises the bar a little, but scientists are used to being surprised.  If it contradicts something previously proved, that raises the bar a lot.  It puts on you the burden of not only showing that your work is right, but showing specifically why the previously accepted work was actually wrong.  (You can't just say "Mine is right, therefore theirs must be wrong.")  You didn't say anything about having done that.  (And if you can't find a flaw in the previous work, then your claim is in fact along the lines of "Mathematics is inconsistent".  The bar on that one is more or less on the moon.)

I also left a similar answer, even up to the (independently chosen) phrasing "moving mountains" versus "moving heaven and earth".  I should say though that getting the community to study your revolutionary work is a good way to go and common, but I don't think it always happens this way, especially among people that have few ties to qualified experts.  I brought up the (amazing) example of Yitang Zhang in my answer.  So far as I know, he really did work in isolation and submit his paper to the Annals rather than shop it around much.

@PeteL.Clark: Heh.  Great minds, etc.  Thanks for your nice answer, the example of Zhang, and especially the deconstruction of the question.

@NateEldredge "The bar on [proving mathematics inconsistent] is more or less on the moon" Note that mathematics can't be proved consistent, so in fact the belief that mathematics is consistent is also just intuition :)   Of course, the community isn't accustomed to being that surprised!

Your answer reminds me of a really interesting story.  Short version: serious mathematician thinks he's found a proof that math (PA) is inconsistent, explains it clearly enough that experts can understand, a top mathematician reads the outline and spots the error, and the author retracts the proof.  So even really extraordinary claims will be looked it if they're explained clearly and reasonably by a sensible person.

@IstvanChung well, technically, isn't it consistent by definition?

@Lohoris See Gödel's Second Incompleteness Theorem.

It might be possible to prove mathematics consistent but only if it is inconsistent

@NoahSnyder: Late comment, but I've been trying to track down that case/link for months/years. Thank goodness you included it here!

By attacking your own proof even stronger than the others do.
Seriously, there is a reason why people in your discipline haven't been able to find the answer for centuries. The a priori probability that you are wrong is so high that even when you have created a good looking proof, the a posteriori probability that you are right is way too low. This means that, if you know enough of your own discipline, you should not be convinced that you solved it. Given a problem which has resisted solution for a long time, being convinced that you solved it just because you have a proof you believe in is a sure sign of a crackpot.
So non-crackpot behavior in such a case will be to try to take the proof apart, shoot it down, tear it to pieces from all possible angles. This is what your peers will be doing, and this is what they will expect you to be doing. To forget your pride, your subjective biases, and to be merciless to your own result.
They you only believe you after you have found more ways to disprove your result than they themselves can think of, tried them all, and failed in all of them. And your paper has to clearly show that this is what you did. Anything else will earn you the crackpot title.

May I add to Nate Eldredge's  comprehensive answer that, if your work shakes or shatters the commonly held views in your community, then it is very important that you reconcile those views with yours, by which I mean: show exactly where the community is "wrong" or "not exactly right" and why. Offer counterexamples, predictions, all you can.
Relativity would be nowhere if it didn't reduce to good old Newtonean mechanics where the latter performed perfectly!

And sometimes it's appropriate to explain why it seemed that what's been done couldn't be done.

Despite what people will say it is true that journals will reject papers using author profiling without a proper review. It is hard to say how many papers are rejected this way but Elsevier say that they reject 30 to 50 percent of papers without a review for other "technical reasons". See also this paper about how editors can save time by looking at author attributes such as affiliation to reject papers without looking at them.
I have personal experience of this because I recently made significant progress on a well known 100 year old open problem after experts in the field had said that future progress was likely to be very slow. The journal I submitted the paper to rejected it as soon as I confirmed that I had no affiliation. There was no reviewer report and they did not give any specific reason. I had complied with all their technical requirements for submission.
However, I pointed out to them that according to the code of conduct of the committee on publication ethics to which the journal claims to adhere "Editorial decisions should not be affected by the origins of the manuscript" and "Journals should have a declared mechanism for authors to appeal against editorial decisions." To my surprise they responded after a delay to tell me that they would look at it again.
It is true that there are many claimed resolutions of problems such as P vs NP that can be dismissed at a moments glance. This can be done because there are well understood reasons why these problem are hard and a solution would need to address that. Many claimed proofs of open problems by non-academics descend quickly into non-standard language that makes it hard to even address why they are wrong so they are just ignored by the community. It is up to the authors to make sure they communicate their ideas correctly.
If you do have a solution to an open problem my advice is to submit to an open repository such as arXiv. If you can't get an endorser use viXra or figshare (full disclosure: I am viXra admin) Do not pay attention to negative things said about viXra. It's purpose is just to give you an independent time-stamp and an archived copy you can point to. It does not attempt to review or give your work credibility in any way. The last thing you should do is submit to journals or send to experts without having a verified public copy because if it really is a breakthrough there is a real risk of plagiarism that can only be averted by having a prior copy archived.

Your first paragraph seems to indicate that the link is to an Elsevier editor admitting that they use author profiling to reject papers, but there is no such thing at that link (he lists perfectly valid technical reasons to reject a paper). Also note that in the mathematical community, putting something on viXra will mark you as less serious (whether this should be the case or not), so I would not advice anyone to upload anything there unless they really have no other options.

My first paragraph does not say what you thought. You are reading things between the lines that are not there. If you claim a proof of a well-known problem as a non-affiliated author you will already have problems being taken seriously and will find it very hard to upload to arXiv. Even if you have an endorser arXiv will always hold a non-affiliated submission for days pending moderator review and may then reject it exposing the author to possible plagiarism. viXra will upload very quickly without such moderation. figshare also seems to work quickly if you prefer.

I am fully aware of what the first line says. I am mentioning to you what it might be seen as saying to someone reading it for the first time.

I will modify it try and make it clearer

fyi based on his profile PG lists his website as Vixra Blog seems to be affiliated with the Vixra site, an alternative to arXiv, which is undisclosed in the answer

You need to read the answer again where it says "full disclosure: I am viXra admin" Note also that it is not a commercial site. We have no need to tout for business. I mention it along with an alternative only because it is a relevant part of the answer.

"if it really is a breakthrough there is a real risk of plagiarism". This is the sort of thing that often gets people identified as cranks - they believe that editors and reviewers are going to steal their work at the same time as rejecting it as being invalid. Do you really believe this is a risk when submitting to reputable journals, and are you aware of any cases where it has happened?

Yes I believe there is a risk and even if it is a small one it is something that authors should take steps to avoid. I am not aware of any identifiable cases but that does not take away the possibility and risk of it happening. Plagiarism in academia is getting so blatant that you can now find papers that apparently have been copied almost in entirety word-for-word three times over see  http://academia.stackexchange.com/questions/18512/how-can-i-find-the-original-article-of-an-often-plagiarised-work It would be naive to think that this is not the tip of the iceberg.

Worse: papers are accepted because of author profiling. That said, if you have to select 20 papers out of an ever-growing number of submissions (people in some areas have come to publish like rabbits procreate) I can understand why you would pick 50 papers you assume to be good by some criterion and have only them reviewed. Understand, not like.

Yes it is understandable if editors want to save time but it would be dishonest to do it and then claim that they adhere to the COPE Code of Conduct.

@PhilipGibbs The post you link is not about plagiarism by reviewers or editors of reputable journals who had rejected the material and then passed it off as their own. So, again, are you aware of any cases where that has happened?

David, the risk of it happening exists independently of whether or not it has ever happened or been documented to happen. The advice I am giving to anyone who makes a discovery is to not take that chance if they value the priority of their work.

@PhilipGibbs Of course. Likewise, the risk also exists that space aliens could steal your work and use it to produce a death-ray that kills us all, independent of the fact that this has never happened. The question is whether these risks are large enough to be worth worrying about. The magnitude of the risks is not independent of whether or not they have ever happened.

I have found a couple of case files where reviewers were accused of plagiarizing work they were supposed to be reviewing. The conclusion in both cases was that it was just a failure to declare a conflict of interest when they were working on similar research. In one case ( http://publicationethics.org/case/parallels-between-unpublished-manuscript-and-published-article-other-authors ) the paper was rejected by the reviewer and in the other ( http://publicationethics.org/case/reviewerauthor-conflict-interest ) he had simply not done the review.

David, If you think the risks are not high that is fine for you. I would not take the chance.

@jwg This risk does exist and it does not help ridiculing it. See Jocelyn Bell and Rosalind Franklin. As student our group found out that one assistant used the data of another assistant without permission. Mark Chu-Carroll from "Good Math, Bad Math" had a bad encounter when he talked to another person about a new idea and found later out that this person put out a paper with this idea without acknowledgment.

@ThorstenS., as David Richerby pointed out, listing various plagiarism cases is not the same as citing a single case when reviewers or editors have rejected submitted material only to plagiarize it. I did not 'ridicule' this - I pointed out, I hope helpfully, that this kind of claim is exactly what people look out for to recognize cranks. If you are submitting serious work, you are doing yourself a disservice by appearing cranky. On the other hand, if your submitted work genuinely has been plagiarized, you should make a fuss about it.

@jwg Well, here we go: The American authors Vijay R.Soman and Philip Felig from the University Yale did exactly that: Rejecting a paper from a NIH group and stealing their data for their own purpose. The German cancer professors Herrmann and Brach rejected a grant wish from a Netherland group, translated the original arguments from Dutch to German and asked with this for their own grant.

@ThorstenS. I actually find it quite comforting that the best example seems to be one from more than 30 years ago, and which is not even quite the reviewer plagiarizing (rather, he failed to report a possible conflict of interest and showed the paper under review to a subordinate who then plagiarized it).

@TobiasKildetoft Erm, Herrmann/Brach is from 2003. In advance: I do think that most scientists are honest. But apart from the fact that the goalpost is now moved (10/100/1000 cases ?), I think I should mention that universities could be sued to pay back grants completely if fraud is detected. Another interesting fact could be that frauds (like Jan-Hendrik Schön) are mostly caught by anonymous tip-offs which may have some reason because Reiner Protsch destroyed the careers of several people who tried to warn other scientists.

@ThorstenS. Right, but that was not a reviewer of a new result, that was a slightly different situation.

I have had to do both with cranks and with true plagiarism cases - both exist. The fact that, if you are lucky, you won't have a traffic accident does not mean you should be careless. I can sympathise with someone who believes they have a groundbreaking result wanting to protect their priority (see also Perelman vs. Chern where even being seen as competent and the relevant paper being out there early enough did not prevent a priority dispute). Unaffiliated or weakly affiliated researchers are particularly vulnerable. I have my doubts about cranks, but let's not dismiss the plagiarism danger.

@CaptainEmacs Sorry, there seems no Perelman vs Chern (who died in 2004), and the reason of Perelman's seclusion seems to be unjustified (I don't totally believe the reports on The New Yorker).

@Yai0Phah Sorry for the delay, I don't seem to have been notified, or I overlooked it. I mentioned Perelman vs. Chern because this is well-publicized. If you say that the New Yorker is not reporting correctly (do you happen to have sources for that criticism?), then I have also some cases from my own experience and that of colleagues of  attempted, partial and completed plagiarism.

@CaptainEmacs As I said, there is no known conflict between Chern and Perelman. I did not say whether New Yorker reported correct or not - I said that there is no confirmation of the verity, since Perelman has never, as far as I know, openly made any comment, and some mathematicians complained that they were incorrectly quoted. Furthermore, Perelman was neither "unaffiliated", nor "weakly affiliated", and his paper was uploaded to arXiv. In mathematics, there are very few breakthroughs owing to unaffiliated, while crankery is a much more common phenomenon.

@Yai0Phah I agree about the crankery. My point was about priority disputes and underhanded scooping. Unfortunately, I know that it happened in the case of friends, and I was involved in a case that required intervention to prevent this from happening. These are rare cases, but they are far from impossible and do happen.

@CaptainEmacs Priority or credit disputes are much more complicated than blunt plagiarism. At least in mathematics, uploading to arXiv seems to be conventional and this does prevent blunt plagiarism. However, it does not prevent "stealing ideas", and there are still issues much subtler (for example, completing a "non-rigorous" or "incomplete" proof, let alone independent developments). Perelman's case is far from being a blunt plagiarism anyway.

@CaptainEmacs were you thinking of Yau, not Chern?

@DavidRoberts I forgot the details of the case, and don't have time to check now, but I believe to remember that Chern was, in some way, involved. I have no opinion about the case, only that it appeared very unpleasant.

@Yai0Phah I mentioned cases from my own range of observations. These were not pure priority disputes. They cover whole spectrum of unpleasantries discussed above. It's definitely not merely theoretical. If it were a single case, I might dismiss it as personal anecdote, but there were multiple instances of different types and levels of such events. As I said, not very common, but they do happen. A bit like fires. They are rare, but they do happen often enough to require fire stations.

Some advice is to very carefully check that the proofs are correct, ask one's supervisor for advice, and seek third opinions. Perhaps the supervisor has colleagues in the research area who would be willing to read the draft and offer concrete feedback.
If the journal submitted to is good, yet the reviewers did not give any useful feedback at all, then there is almost certainly a problem with the abstract and introduction.
The abstract and introduction should make clear the new idea that allows this "breakthrough". Presumably many have approached this problem in the past and failed; there may be widespread beliefs about why it is difficult to prove or perhaps even known "barriers" to attempted proofs. The abstract and introduction should clearly and briefly mention why such beliefs, objections, or barriers do not apply or how they were overcome.
In short, the abstract and introduction must give the skeptical reader reason to believe the paper could be correct, given the reader's background knowledge. If this is done, I would hope that reviewers would at least mention why they do not believe the result.

While most of the answers seems to have much confidence in the academic system, I would like to offer another viewpoint.
I think it is in fact much harder for an unknown (to a specific field) to
present a solution to the scientic community than normally expected.
Scientists do screw up and sometimes royally.
First example: The infamous Monty Hall problem.
More than 65% of all professional answers to Marilyn (with all sorts of academic
grades including statisticians) strongly rejected their answers, sometimes with
outright jeers and taunts.
This included Paul Erdos and Straight Dope Cecil Adams.
So even the majority of experts can fail.
Second example: The also infamous neutrino anomaly.
The interesting thing here is not the error itself, but the reaction on
Arxiv. Anyone who would have dared to offer superluminal theories before
the announcement would have been immediately declared as relativity crank.
After the announcement papers came flooding in offering all sorts of
superluminal theories explaining what we know now to be simply a
bad cable.
What are the problems an unknown may face ?

Arxiv. You need an affiliation from an university or research institute
and/or an endorsement from a know author. Arxiv can revoke or limit your
access without explanation. This requirement also applies to fully qualified
scientists which are working in companies.
Journals. Too many people are trying to get their results published in too few
respectable journals. Journals also pretty scale bad, you have to wait a long
time to get published. Lesser known journals may have lower barriers, but you
have the real danger that the contribution is missed. And even the lower
journals may reject the paper.
Scientists. The situation is different in various countries, but normally
scientists are overworked and underpaid. They have not the time nor the
resources to review contributions with the very slim chance to get a
scientific jackpot.

If someone thinks the viewpoint is not valid, try just for fun to supply
a normal paper under a pseudonym and the home address.
The only viable option I see is to get contact to the scientists in the field
and try to work with them over the contribution which may be harder than it
sounds. The list provided by Kaveh is a good resource to start with.

Your assessment of the Monty Hall issue is incorrect. The underlying issue was that the problem was not well-specified. The subtle ambiguity in the specification was what caused the disagreement between experts.

@EnergyNumbers Yes, I heard that excuse. But the original question does not offer any ambiguity. The host knows the position of the car (it is not random), it is a gameshow (he cannot open the car door) and he chooses another door with a blank. And, really, what allmost all critics said was not:"It has two different solutions, the answer depends on the following circumstances etc. pp.". What they mostly said was: "THE CHANCE IS ALWAYS 1/2, YOU STUPID FOOL !!" Both Massimo Piattelli-Palmarini and Gero von Randow, a science journalist, explained the solution and the people still disagreed.

To stop just a discussion in progress: My argument is simply that scientific consensus can screw up royally. If you are not believing that this can happen, look up some of the original quotations prominent scientists did about Wegener's continental drift and Fritz Zwicky's dark matter at the time of the discovery. Believe me, you do not want to read this.

The neutrino result was a very special case. It was announced very, very cautiously, almost as "Er, guys? We have this kinda crazy result that we can't explain. We tried for ages and we found mistakes and fixed them and still got the same thing. Can you guys see what we did wrong?" With that amount of care taken in the experimental work that seemed to maybe show faster-than-light neutrinos, it becomes worth considering. Most FTL results are just some guy saying, "Yeah, but what if Einstein were wrong and..." or, worse, "Einstein says this is impossible but, if we subtly redefine this, then..."

@DavidRicherby And because they asked what they did wrong, people should wait and send proposals instead of submitting FTL papers. Violating relativity is a really terrible problem. All particles have "normal" rest mass (m=0 => v=c / m > 0 => v < c), there is no way to get a non conflicting description of FTL events with separate observers because RT is needed to describe time and space properties correctly. Please explain to a crank that FTL of normal matter should be considered impossible after seeing how the "experts" were willing to accept possible FTL and even had proposals to offer.

@DavidRicherby When I heard about the "FTL" neutrinos, my first question was "is the sensitivity high enough"? Because it is well known that group velocity can overtake front velocity and when the thresholds are not sensitive enough, this can create the impression of faster-than-causal-velocity processes in experiments. I had no idea about the detailed setup, but this would have been my first question.

From my experience as a graduate student, it seems that researchers who are passionate about their work do approximately the same thing regardless of its supposed importance.
Here's what to do:
(1) Write up your work as best you can while also discussing the ideas and concepts with your friends, collaborators, co-workers, etc.
(2) Promote your work to other researches, friends, collaborators, etc.  But, often people are busy so keep your expectations low and courtesy high.
(3) Pinpoint your audience and seek out the appropriate journals, workshops, and conferences to submit to.
(4) If you have any questions or concerns before submitting, contact someone affiliated with the conference.  I've had a positive experience doing this, but sometimes you get ignored.
(5) Be ready for rejection because you are more likely than not to get rejected (even if your results are important, actually especially if they are important).
Remarks:
(a) I like to think that the quality of reviews is associated with the clarity of the paper, but I've received some questionable reviews in the past that seem like scattered words that may or may not be related to my work.  Just don't hold a grudge and happily try again.
(b) If something's important to you and you're financially capable, then getting rejected once or twice is alright as long as you keep trying to improve your presentation and keep communicating with others.
(c) You don't want to go off the face of the earth and live like a hermit.  That's not going to help anyone and especially not you.
(d) Finally, be open minded.  People do make mistakes and sometimes the thing that caused the mistake is meaningful.

Some very important papers have been rejected first, some were not even published.
If you are sufficiently sure about your result, and want to set a date for your discovery, and are not afraid that it sometimes takes time to demonstrate and convince people  of the correctness, put your paper online on some open archive, while you work at an hopefully published version.
Some readers may discover a flaw, and perhaps help you publish.

First of all, "solving famous open problems" does not happen in a vacuum. There must be a good reason why the solution occurred to you after having eluded many others.
A possible reason is that you are an expert in some new technique or method of analysis. Then the trick is to establish yourself as an expert in this "new" field. Once you've done that, it is easier to claim that your mastery of this one area enabled you to solve the "unsolved" problem (providing you can demonstrate the relevance of your field). For instance, if you were a pioneer in subatomic physics who discovered that Newtonian physics didn't work in the subatomic area, acceptance of your "proof" would hinge on people's acceptance of you as a subatomic expert.
The other thing is if you have really discovered a new solution to a problem, the implication is that a lot of what is currently written in the field in relation to this problem is wrong, or at least needs to be re-thought. The way to prove yourself is to start identifying at a low level, and rising to progressively higher levels, applications that are now "voided" by your discovery. If you can prove that a whole "stream" of ideas needs to be re-thought, and then present your discovery as a "common" solution, people will take you much more seriously. An example was when people figured out that you could create a new system of "non-Euclidean" geometry just by changing a few assumptions.

Probably the single best criterion here: "There must be a good reason why the solution occurred to you after having eluded many others." +100!

Prepare for your fearlessness. You will need to use it frequently.

Let's say our goal is to accomplish Kaveh's list. Assuming he has some experience in academic research, then each of these statements are equivalent:

His priority is not really to have his work accepted, but actually about not being misunderstood (the actual premise of the question is about having productive feedback)
For him she is just stating the obvious (the upper part of the list), and telling him to do things he is already doing (the lower part)
She needs him to accomplish the list, but he cannot do that if she accuses him for doing the thing he didn't do
The problem of convincing her that he is not a crank is reduced to the emotional and misunderstanding problems

I think that in all discussions about cranks the overwhelming assumption is that he is hubris. This is, unfortunately, one-sided, because the actual emotions should be guilt and frustration . He is guilty for worrying he is hubris, and frustrated for her misjudge that he is hubris. With all my respect, think most experts lack the necessary experience to give an efficient advice.
 Because this answer doesn't attempt to answer the problems that most established experts want to address, it is basically a frame challenge answer.
The rest of the answer is just an elaboration on how to deal with emotional reactions and misunderstandings in each specific case. Here is the table of content for this answer:

Dealing with your emotions
Dealing with their reactions
Other problems
For those who want to help
Links

You can also read an answer for a generalized question (what challenges one may face during the project, not just how to convince the experts). I will assume that your platform to get feedback on your work is Reddit, but it's not a big deal.
Dealing with your emotions
First, there are two things to begin with:

Epistemologically, it seems that in order to solve a hard problem, you need to be aware to the solution a priori
Epistemology is a field in philosophy studying about the nature of knowledge and how we acquire it. I haven't read much about this though.
Psychologically, when you are aware of yourself, these self-conscious emotions will evoke: pride, shame, guilt, and embarrassment

Because doing research is to prove that you are wrong before proving you are right, a natural thing when you think you know something is have skepticism about yourself. So alongside with the specific problem you are working with you will ask these self-conscious questions:

Am I claiming a thing that I haven't proved it yet?
Am I claiming that I know better than the experts, when I lack formal education?

Since the answers to those questions should be yes, they will evoke self-conscious emotions, drifting you from the one and only thing you should focus on: the problem you are working at. You will enter a feedback loop of questioning yourself: the answer for those questions is the reason you questioning yourself at the first place. The real answer lies in the work you are researching, not about whether you are sane or not.
To deal with fantasy (a kind of pride), try to imagine how a life of a famous person would actually be. For every kind of success you are dreaming of, there are people who have already got it. So say you dream about the Nobel prize, ask yourself this question: how did Einstein live with his fame and money? When you realize that in fact all famous persons are annoyed for being famous, then your fantasy is cut. By being able to put yourself into their shoes, you can detach to your emotions, and get back to reality. At that time, if you still wonder what you will do when you become famous, then the only thing you want is a dark and quiet place to have a dreamless sleep through it.
Please be aware that you may have messiah complex. You may not have grandiose delusion, but the thinking that (1) only you can provide a solution that no one seems to see, and (2) in other to deliver that solution, you have to overcome the skepticism regardless how harsh it is, can develop this complex. People may also accuse you for performing gaslighting as well. All of these will make the guilt about yourself even stronger.
To distinguish them with true psychological problems, whose root is insecurity, I will call them as "intellectual messiah complex" and "intellectual gaslighting".
Because the fear that you are delusional only continue when in fact you see that you still have fantasy, so when you have successfully cut it, then the fear itself will disappear. You will not feel shame, guilt or embarrass about your work anymore.
In general, the emotions are only intense when your research is still at the vague phase. The more knowledge you acquire, the less frequent they show up.
Dealing with their reactions
In my experience, there are these types of unproductive response: labeling and sarcasm.
Labeling
When they tell that you are crank, just give them this crackpot index, list all the points that may apply to you, and explain how they are wrong. If they accuse you for being arrogant, tell them that confidently walking into a storm with a smile is different to seeking for attention. If they think that you are insane (walking into a storm with a smile is insane obviously), it would be much easier if you can have the conversation face-to-face. Only by seeing how your eyes are determinate but your mind is not closed at all, that they can assume that you are not. Don't be embarrass for telling them how you have prepared for the project.
Let's discuss more about insanity. I think it is best to "help" them reach the conclusion that you are insane, because it is no longer a label on your behavior, but a label on your rationality. When you reach that stage, they will have a strong motivation to continue the conversation, and your evidences will be listened carefully. If you can create a cognitive dissonance in their mind, then their pattern-matching will be silent, and they will not be locked in their perspective anymore.
Respond to their labelling (crank, arrogant, insane, stupid, high, bias, spam, word salad, not even wrong, woowoo, pseudoscience, time- wasting, etc.) by giving the definition of the word and show why it doesn't apply to you. Have a note listing all your prepared answers for each label, so that you don't need to rely on your poor memory. To deal with skepticism, you need to immediately form perfect combination of words, and your brain will put you down. (More details later.)
Sarcasm
If they simply making some jokes, it may be true that it is actually funny. In that case, perhaps it is best to continue the joke. Seeing how you actually enjoy their jokes will make them see that it doesn't affect you at all. Like if you are a fat person, then joking on your fatness will make them see that you are aware of your shortcoming, and it's just that you don't have time to solve it yet. If you are a blind, then having a joke involving your disability will make the non-blinds astonishingly surprised. They will perceived you as invincible now.
If it's really hurt and you can't think of a clever thing to say, then perhaps it's best to remind them that they are in a place that mocking is not acceptable. Like a smack down show, they literally see you as the arrogant prideful guy who need to be taught. Note down links to comedic shows, where making fun of others is the goal, like WWE shows, higaniga conspiracy theory videos, or mocking subreddits, and show them how hilarious their actions are. For example, you can say:

Wow, r/Buddhism becomes r/WWE now?

Or you can invite an authority to deal with this:

I don't think this behavior is appropriate. I'll report the mods/I believe others don't find this appropriate as well.

In short, be clever when dealing with ad hominem. The problem is that you can't be clever when your mind is clouded with anger. So you need to prepare for clever moments. When you can do that, then their next response will likely be productive again.
But in all, don't interact with them because you feel misunderstood, but because of the knowledge they have. Try to convert questions about your identity/self (e.g. "am I crank?") to question about definition (e.g. "what is crank?"). Show how you possess a rigid body of knowledge of what they are trying to convey, with phrases like "psychologically speaking", "what you are talking about is called ______ in philosophy", "in the field of informal logic", etc. They are relevant interesting academic fields that they don't know of. If you think that they have knowledge on what you need, only focus on that, and ignore any unreasonable critique at you.
When you have solved all of the emotional problems, then 90% of the conversations now are productive. But there will still other problems.
Other problems
Memorial problem
You may have wasted so much time to read about crackpotism, conspiracy theories, delusion, etc. You may have wasted so much time on conversations that at first people strongly disagree with you, but after some talks it turns out that they don't really disagree with what you actual mean, and all of this is just misunderstanding. Take note on every evidence that that make you feel you are on the right track. Don't let your research about your sanity gone.
The problem of being unable to explain yourself is because of tacit knowledge and the tip of the tongue phenomenon. When you have a very strong evidence that you are not a crank, then after a week or so what is left is just a feeling that you are not. Lacking the evidence, you will be dragged into the loop again. You need to immediately make a perfect combination of words to get out of this.
Misunderstanding
Even when miracle happens (they spend their time and energy to analyze what you say), there will be a very strange phenomenon that you just can't understand: you always feel that they implicitly agree with what you say, but you two can't settle to a consensus, and thus just going circular.
I think, all conflicts in the world come down to the problem of priority. Usually the situation is like this: person 1 can see that problem A is more important than problem B, and person 2 see that B is more important than A. The problem is that, most of the time both A and B have to be solved together, or else none of them can be achieved. But because both insist that their point is more important, both will miss and pass each other point. Both will feel the conversation is unproductive, and sooner or later one will drop it.
When this happens, it is just a blind leading a blind, or worse, a blind fight. Investigating the nature of this phenomenon and how to deal with it is my research interest. My advice to deal with misunderstanding is to use negation, not explanation.
Egocentricism
Egocentrism is not about being selfish or have a huge ego, but about being unable to differentiate your mind with others' minds. When you find something interesting, then you will automatically assign that others will find it interesting as well, and will be confused when in fact they don't really care. However you remind yourself about this, this tendency will still activate.
Don't assume that they will assume that their feedback is wrong. You may be open minded, and they are too, but in practice both of you can't. If you assume that they are curious to know why they are wrong (a kind of trust), then you will tend to give explanation. But in fact they just see that you are defensive. They accuse you for things you have never done, and then either passively drop the conversation or actively block you from further explanation. I term this as "intellectual silent treatment".
Seeking feedback
When an idea pop up and you feel wonderful, your gut will still tell you that there are much more fields you need to read carefully. Although you don't mind spending more effort on researching, you just want to ask for feedback because it would be much more efficient. Your egocentricism assumes that people will get what you get too. If you want to finish the project as soon as possible to move on to other important things, then your urge to share it will be higher. But STOP! Posting it now will only receive harsh, unreasonable critiques. Listen to your gut, and read all the fields you need first. Good questions only come when your mind is in the ignorance stage, which is the result of understanding the field as it is.
Having said that, at some point you will see that there is no point to ask questions, and you just want them to read your work as it is. If you have the necessary requirements (literary review, inline citation, methodology, etc.), then you can now submit to an academic journal, and you don't have to be afraid for being misunderstood as a crank anymore.
Receiving feedback
In case your first and foremost audience of your work is the popular audience, making your writing style necessarily un-academic, then the situation will be complicated, because your most important audience is the academic one. (Yes, sometimes the important thing is not the one you should prioritize – see the Eisenhower method.) Because your work has to serve two different kinds of audience, who have different background, knowledge and expectation, you will have mixed feelings when receiving their feedback:

Popular audience cannot provide useful feedback, but their excitements indicate that you have touched a big problem that they are looking for
More knowledgeable readers or even academics from distant fields can give useful knowledge, and can play the role of initial gatekeepers. But once they say good luck to you, you know that they cannot help you anymore
Academics from relevant fields will feel it's vague or wishy washy, because they expect your work to truly be presented in academic form. But if you explicitly say that this is just the phase to capture what you have in your mind, then their attitude will flip 180 degrees

Let's talk more about the last point. In your mind, having no literary review, methodology or dataset is not important, because you already accept that you don't have one. Therefore, you must say explicitly that the article is just a cursory research to sketch a roadmap for your study, and list all of your shortcomings as best as you can remember. Without this part their expectation will lock their minds, and any of your explanation from now on will be perceived as defensive.
For those who want to help
Here is my advice for those who want to help. Hopefully it can reduce wasted effort, and bring you maximum happiness:

Always assume that the person you are talking at has something interesting that you can learn
Be conscious that although they may not know what they are talking about, you may not know what they really want to convey either
Confirm their correct observations before address your concerns
Use Socratic questioning

Don't feel threaten when they show signs of crankiness. The fact that they accept to be labeled as crank indicates that they have something more important to do. Like you, they are rational creatures, and they have already run cost-benefit analysis before starting the project. When you want to give them advices, you may want to frame it like this:

I don't think there is another option to do X / The best way to do this is Y. But since it seems that you know this too, can you explain why you don't think it suits you?
I don't think you understand concept Z as it is. In my opinion, Z is about a, b, or c, and perhaps c is closer to what you mean. Is that correct?

You may want to learn about conceptual metaphors if you want to know why sometimes serious ideas are hilariously crazy. I recommend the book Metaphors we live by by Lakoff and Johnson.
Links
Kletische has some good articles about this:

How to Handle False Accusations in Everyday Life
Why You Win When You Forgive
How to Handle, and Grow From, Hateful Criticism

You can also read my research: A theory of perspective. It discusses about various things, two of which are intellectual betrayal, and cold gaze, which are relevant to this answer.

What an absurdly long answer. And yet it seems to hardly contain anything of use to the OP, being mostly meaningless.

the other answers assume that the researcher can keep a cool mind thoroughly, and the experts don't give a single mock on the first ideas, but just either give feedback or remain silence. This is not the case. If possible, can you elaborate why the first two sections are meaningless?

think about it, maybe only the first two sections directly related to the question (how to convince). The other points are more about a generalized one (what challenges one may face). I'll removed them then

This does not answer the question. This answer describes how OP should deal with their own mental/emotional health (which is important, but the stated advice is off), and lectures the research community about how they should treat outsiders (which is tilting at windmills), neither of which address OP's question of how they can best convince other people that their solution is legitimate.

When they tell that you are crank, just give them this crackpot index, list all the points that may apply to you, and explain how they are wrong. — This may be the single least effective way of convincing someone that you are not a crank that I have ever read.

@JeffE It turns out that this answer is a frame challenge answer, in which a seemingly off-topic advice is necessary. I add the explanation on this at the top, can you check it?

Sorry, no.  I would find the frame challenge appropriate if OP had asked "I have been labeled a crank; how do I convince people that I'm not?" or "How do I deal with ad hominem attacks on my research?", but that's not what they asked.  (And the advice that you offer is still off.)

@JeffE What is the difference between your first example question and the titular question? And in any case, what kind of question that you think this answer is appropriate and good?

The difference is that OP doesn't say he's been called a crank or disparaged in any way.  They're trying to prevent such attacks in the first place, not recover from them.

@JeffE Trying to prevent it will lead to the self-fulfilling prophecy. When a novice claims that he has a big idea, then because of his lack of jargon, he has to use analogy to express it. This will make the idea so absurd and funny to the experts, so sooner or later he will be laughed. Because of that, it's better to expect ad hominem attack and equip the skills to deal with it, rather than try preventing it with all cost and be disappointed when it comes.

You just answered the question: Follow the language and customs of the audience you want to convince. If you don't know the language, don't just use "analogies" — learn the language. Read. Study. Listen. Be patient.

@JeffE It's complicate, but I can assure you that being patient is simply impossible. It's like the frustration that you don't want to be infidelity, but you still have erection on sexy girls. I think for a novice to (1) actually see something that experts in the field haven't seen, and (2) haven't read textbooks yet, the only way this is possible is that (A) it's an interdisciplinary research, and (B) it bases on an analogy. Both A and B make both the researcher and the expert don't even know what fields he should look at, and thus both assume that her field is the correct field.

To not assume that he is talking about her field, and thus not feel he is insulting her field, she has to know the field he should read first.

I can assure you that being patient is simply impossible. It's like the frustration that you don't want to be infidelity, but you still have erection on sexy girls. — And I can assure you that patience is not only possible but common. It's like being aroused around sexy girls and then not assaulting them.

@JeffE You can only stay calm if you have a trust that your love to your partner is true, or even irreplaceable. If, for any reason, there is a crisis that makes you doubt your love to her, despite the fact that you actually does, then any arousal you get will add more to your existing anxiety. Calmness/patience cannot be achieved when feeling guilty. By "impossible", I mean the cognitive load. If we take reading textbooks as our goal, then I just want to examine what factors can impede it.

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text:
A professor matches my previous research and the other matches my interest. Which one should I choose?
I am a fresh PhD student and I need to choose an advisor to guide my research. Currently I find it difficult to choose.
One professor's current project pretty well matches my previous research experience. Although I would not say I am interested in it, at least I don't hate it. But I find most of his students will need 7 years to graduate the PhD program under his advisement. The speed of this professor's life is really slow. The most important thing is that he rarely gives his students constructive suggestions when they confront problems in the research. I know a PhD needs to have the ability to conduct experiments independently, but some kind of guidance is necessary.
As for the other professor, his project is really exciting and he published many papers with high impact factors. It is not that related to my previous experience but it may use some of the techniques I used before. I like it but some of my friends told me this professor is really picky and tough. With him as an advisor it may be difficult to get job after graduation.
Currently I am working with the first professor. I try my best to make myself get interested in the work, but I am still hesitant.
What advisor should I choose to achieve a satisfying research experience?

some of my friends told me this professor is really picky and tough — That one.  You want that one.

I try my best to make myself get interested in the work, but I am still hesitant. If you are having a hard time being interested now, just give it 3 to 4 years, and imagine how much worse it will be. Especially after this work you aren't even that interested in has been rejected from multiple venues of publication, you are working late nights revising the papers you aren't even that interested in instead of going out and socializing, etc.

I just realized this is a few years old. What did you do and how did it turn out?

@mikeazo, slightly more than a year, really...

@mikeazo, misread the date :-(

Ultimately, I would choose to work on what I'm interested in rather than what I have experience in, assuming that you have enough background to follow your interests (but in a 7 year PhD program, you'll have time to learn the techniques you'll need) and that there is a future for you after completing the program.
(Learning how to use capital letters and to punctuate your sentences properly will also help in the future.)

Always, always, always pick the advisor who is less of a jerk.  Doing so serves numerous purposes:

The project will be more enjoyable, because you're not working with a jerk.
If the initial project doesn't work out it will be easier to switch to a new topic, because you're not working with a jerk.
Even if the whole program doesn't work out, it will be easier to get support for switching labs/schools/careers, because you're not working with a jerk.
It means fewer talented students will pick jerks as advisors, which will (hopefully) help weed out jerk advisors.

In short: talk to the students.  Make sure you're not signing up to work with a jerk.

Note, however: "Jerk" does not mean "tough".  You want your advisor to have high standards and to hold you to them.

Yes, agreed. Tough: good. Jerk: bad. @user9983: seems like you have a pretty good handle on the "tough" question.  Now go find out about the "jerk" question!

A Professor here.  In general, you want to go with the Prof. who is research active.  If a supervisor doesn't provide constructive feedback, that simply means he/she doesn't care, has no idea how to do research and don't know the area.  That means you're on your own.  If you're lucky, you'll learn how to swim and churn out passable work.
On the other hand, the 'tough' Prof. produces world class work (based on your description).  This is 'normal'.  Simply look at the best chefs in the world.  Do you think they sacrifice quality?  Their name goes out with every dish/paper!  Same principle applies.  So if you want, and have the capability, to play with the best, then the tough Prof. should be the one.  For sure, it'll be a baptism of fire, but if you're capable, you'll find yourself running with the best.  More importantly, you'll learn why they are the best; their secret: they care very very ... much about their work.

One friend started doing a PhD under a world-renowned star, and gave up something like a year later. He told me he met his advisor twice, once when agreeing to the thesis and once crossed him casually in an aisle. As star he was elsewhere most of the time. He switched to a "second tier" advisor, and was very happy with his decision.

Big profs are like the bugs zapper (device with blue light).  They attract students, and when the student enrols, they get zapped.

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