Your question has some issues. Given some of the questions you have asked on other SE sites in the last few days, I have some reservations about whether your question is being asked in good faith, but taken on its own merits it is a reasonable question so I will try to answer it.

The main issue is that, even in asking this relatively simple question, your writing is far from clear. If you cannot write clearly in this situation, your chances of writing up a difficult piece of mathematics or theoretical computer science are less than good. For instance:

His/Her supervisor(s) accept the work and they published it in a highly known journal and they get rejected.

Laying aside issues of subject/verb agreement and consistency of tense, the entire sentence doesn't make sense: you can't publish a paper and get rejected.

It breaks what most people believe

I don't know what it means to "break what most people believe".

or what they have already proven,

What? Are you saying that your proof contradicts other proven results? Taken literally, that would mean that you have shown mathematics to be inconsistent. In practice this could only mean that if your result is correct then some previously published work is incorrect. If that's the case then you need to be very clear about that and explain the flaws in the earlier work. It distresses me that you don't really seem to believe this but are just throwing it off as loose language.

i.e., He/She solves the P vs. NP problem or any other well known open problem.

Solving an open problem would not "break what people have already proven"....that's what it means for the problem to be open. Also saying "P vs. NP problem or any other well known open problem" is a strange bit of coyness: there is no other problem in theoretical computer science (and very few to none in mathematics as a whole) which is "like" P vs. NP. So it doesn't make sense to give that as an example. It's like saying "i.e., he found the Holy Grail or some other famous cup".

In other questions you have spoken specifically about having a proof of P vs. NP and then upon questioning have retreated from this. This sort of vacillation about what you have done is a red flag of "crankiness" that will make professionals wary.

The reviewers strongly reject his/her work with no justification and they said that the result must be wrong.

Saying that the result must be wrong is not just a justification for rejection, it's the *best* justification. No professional reviewer will say something is wrong lightly. Almost any reviewer who says this will point to at least one specific error. If they do not, then in practice it almost certainly means that the entire document did not make enough sense to them to be more specific.

If your advisor accepts the work, the reviewers reject the work without even explain the mistakes (it is the "best" journal in his/her domain) then what he/she must do?

If you submit a paper to the top journal in your field claiming a solution to the top problem in your field, and your paper does not make sense or does not evince even a correct understanding of the problem, then the editors are likely not to want to spend much time in response. On the other hand, if you are sincerely interested in getting their expertise, it seems reasonable to write back very politely and ask for more specifics about the error. If your response is in any way argumentative then you risk the editorial staff thinking that you will keep hounding them *ad infinitum*, and at some point they *have to* stop replying. So you should write back saying that you are not considering resubmitting the paper to that journal but for your own progress it would be extremely helpful to know what is wrong with it. You could also mention that your supervisor found the paper to be correct.

In fact you could be getting more help on this from your supervisor. If you have really "solved P vs. NP problem or any other well known open problem" and your supervisor believes your solution to be correct, why isn't your supervisor moving heaven and earth to be sure your work is getting the attention it deserves? That doesn't add up. The two possible explanations seem to be (i) your supervisor is being too polite with you: s/he does not actually believe that you have solved P vs. NP; and (ii) your advisor's imprimatur does not carry any weight in the community whatsoever. The latter unfortunately means his/her opinion on the correctness of your work is not worth very much.

A good way to find out whether it's (i), (ii) or -- I do admit that anything is possible! perhaps the top journal in your field is unfairly ignoring your revolutionary work -- is to seek your advisor's help in getting another faculty member to evaluate the work, preferably someone in the department that you can speak to recently.

Finally, you seem to have some real worries that if an unknown person solves a famous problem then it somehow doesn't count. This is really not the way academia works, provided the unknown person is capable of presenting the work in a way which makes sense to the experts (and if not, what a shame, but what else could one possibly expect?). Have you heard of the recent example of Yitang Zhang? Zhang was a non-tenure-track lecturer at the University of New Hampshire when he stunned the mathematical world by proving the existence of bounded prime gaps. He submitted his work to the top mathematical journal...and by all accounts they accepted it *with unusual speed*. In other words, they received a paper from someone they had probably never heard of, looked at it quickly and saw that it was a plausible attack on a huge open problem, and as a result they sprung into action much more rapidly and thoroughly than for most submissions they get. This is an amazing story, but a true one, and it shows how the community responds to a *real* situation like this.

Solving P = NP would break the world. I would proceed differently solving that problem from other big problems. As an aside, if that's not obvious to you, we can safely assume you don't understand what P and NP are. – None – 2016-01-12T04:45:03.050

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

– Michael Lai – 2016-05-19T23:29:12.833@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. – gnasher729 – 2016-09-13T16:18:10.070

@gnasher729 okay, thanks. – None – 2016-09-13T16:26:20.853

1I kind of wish I could give a bounty to @E.P.'s justification for his bounty. Lovely. – Daniel R. Collins – 2016-09-24T16:13:01.223

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. – padawan – 2016-11-10T12:32:05.220

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. – Breeze – 2018-02-13T17:08:41.677

84In that case, the supervisee is

probablywrong 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! – Raphael – 2014-03-24T23:56:39.11018

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.

– Raphael – 2014-03-24T23:58:39.3005I 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

dohave the solution for an important open problem.Butyou 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" -- fordecades. Write forthem. – Raphael – 2014-03-25T00:08:39.28322

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

– Neil Strickland – 2014-03-25T00:12:05.41041) 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 – seteropere – 2014-03-25T02:45:35.907

44@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. – Eric Lippert – 2014-03-25T05:59:57.243

96

"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)

– Joel Reyes Noche – 2014-03-25T07:14:34.3972@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

dohappen 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 – 2014-03-25T08:09:27.3873Please make sure to update this when you publish these results! – user1938107 – 2014-03-25T11:38:04.130

3

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

– Billy Rubina – 2014-03-25T12:19:15.10710@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. – Brian S – 2014-03-25T19:26:32.940

1@Jigg "The proof must be wrong." "You cannot achieve such result." "You do not understand well the notion of ..." – Learning – 2014-03-25T19:42:07.527

7If 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... – Nick S – 2014-03-25T20:50:44.873

6

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

– vzn – 2014-03-25T20:59:50.510can 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) ? – Thorsten S. – 2014-03-25T23:41:25.063

The root problem here is that you've made an unsubstantiated claim, to wit: that you are

nota 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. – Carl Witthoft – 2014-03-26T00:27:49.407@ThorstenS.: Sorry, too many negations for me (at this hour). – Raphael – 2014-03-26T01:35:32.247

@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

doesnot always work well (because of laziness, external pressure and what not) does not mean itshouldnot, 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, becauseeverybodyinvolved is a stakeholder.) – Raphael – 2014-03-26T01:38:42.3809

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

– Rinze Smits – 2014-03-26T03:43:26.17314@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) – vsz – 2014-03-26T13:36:58.533

3Maybe 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... – o0'. – 2014-03-26T13:44:54.647

1

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

– vzn – 2014-03-26T16:17:09.990so hard.@Raphael - Next time you may want to consider posting your comment as an answer instead of a comment on the question. – eykanal – 2014-03-27T02:25:43.333

@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 (?). – Raphael – 2014-03-27T08:17:27.690

14

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 – 2014-03-27T14:36:00.83016@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). – Raphael – 2014-03-27T15:00:40.863

2

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

– Izkata – 2014-03-27T15:26:27.1431@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. – Eric Lippert – 2014-03-27T15:32:41.037

2@Raphael - I'm sure if it worked well enough on enough inputs, it would encourage people to examine his proof more carefully. – mbeckish – 2014-03-27T15:32:45.753

1@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. – mbeckish – 2014-03-27T15:34:15.423

10Just post your paper on the relevant stack exchange. It'll get shot down soon enough. – TheMathemagician – 2014-03-27T16:29:01.027

2

@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

– Kaveh – 2014-03-28T05:10:03.883who haven't read a textbookto read one."It breaks what most people believe or what they have already proven" sounds quite impressive: What problem did you solve? – Marcus Bitzl – 2014-05-13T10:41:53.167