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I have possibly found a somewhat novel method of proving a famous theorem, and after some research, I found a variant of the method published as a paper. So, naturally, I reached out to the professor who wrote that and asked him for opportunities and I am waiting for a reply.

If he declines my request for help on publishing the paper, what other ways is it possible to get a paper published as a person who isn't affiliated with a university/ Has major connections?

Other than that, the process from an outsider's perspective looks a bit tedious right now, but what are the general things to keep in mind while approaching it?

My educational details are that I've passed out my HS this year.

Note: I've checked with some grad students, the result is indeed correct. The current state of whether they'll help me follow through with publishing however is a 'maybe'

Update: Got in touch with the prof and sent the tex file containing the paper to the publisher which he had published too. Hopefully gets some acknowledgment :-)

Update 2.0: Seems to have been done already (See here), not sure if there exists a paper on this but that killed of the novelty.

A word from me to all the answerers: Thank you all. Turns out that my proof was posted before on MSE and I do not wish to publish something already done before. However for the actual question which I had asked, I have received many great answers and I honestly can not objectively choose a single answer which have helped me the most since all of them provided value to me in one way or the other.

As is most relevant to my individual problem of me publishing this result, I will accept Kostya's answer as they were the ones who found that the proof was done before. Again, thank you all.

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    I could have sworn this was asked already, but google-foo fails me. – CGCampbell Nov 24 '20 at 16:28
  • Possible duplicate: academia.stackexchange.com/questions/6191/… – KingLogic Nov 24 '20 at 17:50
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    The most important thing is to get feedback from an experienced mathematician to judge your proof. As a high school student it is difficult for you to see what is noteworthy in the field. There are often many ways to prove a result and it could be that your way is considered obvious and doesn't bring anything new to the table. I am not saying that is the case, but you should keep that option in mind when communicating! E.g. when emailing a professor I wouldn't "request help with publishing" but ask for feedback on whether the result could be noteworthy. You're more likely to receive a reply! – user2705196 Nov 24 '20 at 19:50
  • Noted, sent a reply to the prof – Buraian Nov 24 '20 at 20:31
  • Most research journals want to publish new research, not a variant of something they already published. – GEdgar Nov 25 '20 at 12:00
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Congratulations on finding the proof!

I have looked into your comments and also your MSE post. Here are some remarks:

  • you are mis-attributing the field of your result. It is not 'related to algebra'; modern algebra studies general, abstract features of structures. Neither is it related to operator theory as your tag suggests. If I were to name a field, it would be combinatorics, but the right venue for your proof are indeed high school/recreational math journals, like American Math. Monthly. (Although AMM is quite demanding on the quality.) Serious research-oriented journals are unlikely to be interested.
  • it is not true that posting your solution on MSE diminished your chances to publish it a journal. In Maths, most of papers are submitted to journals after they've been already published as pre-prints on arxiv.org website. For example, here you have a paper in an area similar to yours published in AMM. Posting to arXiv requires an endorsement of your paper from someone who has regularly posted there before, but no peer-review process. Publishing to a journal is then merely a "quality stamp" (of course, if it's AMM, more people will actually read it.)
  • you seem to think that what you need from academics is their "affiliation" and "connections". This is not really the case. What you need is their expertise to decide whether the result can merit a publication, and if so, how to make a convincing case for that. You can see from a linked example above how such articles are written. First, you need to briefly review the history of the problem and known proofs. You then need to compare your new proof with existing ones, and make sure that it is not one of them, and even not one of them in disguise, make the case about it in the paper, and ideally explain what is the advantage of your proof over known ones. (In the case of the paper above, their proof speaks for itself, in that it is just 7 lines long, and relies on an elementary identity whose proof is another 4 lines, so more elementary than existing ones.) You should also place your method of the proof in a context. Is it new? Is it a standard tool to prove identities which somehow was never applied to this particular problem? Are there other problems it can be applied to?

For an editor to give it a serious look, it shoulnd't feel like it's written by a know-nothing amateur who just came up with this proof, typed it and sent to a journal, because in that case, priors are heavily infavor of that proof being either wrong or a (variant of) something known. It must be clear that you have done your homework, know where your result stands in the context, and have a credible claim for novelty.

UPD: as I see now, there is an earlier MSE answer with essentially the same proof as yours. I don't know if there's a published reference, but (a) I would be very surprised if there weren't and (b) it's beside the point, as it would be inappropriate to publish this proof under your own name anyway.

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  • Thank you! I had not seen that. – Buraian Nov 27 '20 at 8:25
  • I asked him for a published reference, but what did you mean that it's in appropriate to publish under my own name? – Buraian Nov 27 '20 at 8:38
  • A scientific article needs to contain an original contribution. If you re-discovered something, it would be a major violation of norms not to cite prior work (regardless of whether it's actually formally published or not) if you are aware of it. And if all you have in your publication has already been done before, why would you write it? Many a mathematician have been there, and usually they just write off losses and move on to the next project. – Kostya_I Nov 27 '20 at 8:56
  • In fact, one way to salvage some output is to write a survey article on the topic. Recently, something like that happened to non less than Terry Tao (with co-authors), when he though to have proved a new identity about eigenvalues, only to have been told that this identity has been already discovered and re-discovered multiple times. What resulted is arxiv.org/pdf/1908.03795.pdf, which surveys prior publications with a claim of originality much scaled down. – Kostya_I Nov 27 '20 at 9:02
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    You should rather ask: how different should it be before it gains novelty? What is it new in you proof that would be of interest to others? It is quite usual that a proof contains many steps, and each of them can be done in several ways. Tweaking any number of them does not produce a new proof, especially if these steps are standard. There must be a new non-trivial idea. – Kostya_I Nov 27 '20 at 9:29
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I'd recommend not trying to submit a paper to a journal without guidance from a more experienced academic. That academic doesn't have to be a professor; anyone with more experience with the field than you should be good.

You've contacted the professor, which is a great start. If they don't reply, you can/should also talk to your teachers. Since you're fresh out of high school, there's a good chance your somewhat novel method might be wrong, or not novel. Your teachers have been in the field longer than you, so they probably know more than you; furthermore they know you personally so they'd be more likely to look at your method. If your teachers are unable to help, you could also ask friends/family for anyone with more formal training in the field.

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  • I've checked with some grad students, it's correct for that matter. – Buraian Nov 23 '20 at 13:44
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    @Buraian you might want to edit that comment into the question, as it is very relevant. – Tommi Nov 23 '20 at 14:27
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Anyone can publish a paper, regardless of age or affiliation, provided that it meets the (rather high) standards of a journal. The standards will include things like understandable writing, but more important is whether the paper solves an "interesting" question in a "novel" way.

"Interesting" can mean new and important, or classic, or other things. "Novel" means that the approach is new, and for mathematics at least, something that might be exploited for solving other problems. A "famous theorem" is itself interesting, of course.

You don't need a professor's help to do this, but they might be able to give you some advice on your paper and how and where to present it.

But even your secondary school math teacher can probably help you with this, provided that they read enough of the literature.

But the way to get started with a publication is to submit it (probably online) to a suitable journal. You will hear from them fairly quickly if it is rejected. If they find it "interesting and novel" it will be assigned to some reviewers for deeper analysis.

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    Do you know of journal's which accept results related to algebra? and how exactly do I submit? – Buraian Nov 23 '20 at 13:12
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    Sorry, no. But I do know that there is a math journal whose mission is to publish work by students. Look at websites of the American Mathematical Society and the Mathematical Association of America to start. Those two organizations (among others) also have more general journals. – Buffy Nov 23 '20 at 13:20
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    Perhaps "may not need" ... most people need a lot of guidance in prepping their first ms (no shame), although I'm not in math – Azor Ahai -him- Nov 23 '20 at 15:34
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A particularly challenging aspect of writing a paper is the introduction. At least in my field of research, I expect this to contain a reasonable overview of related work. Context ist very important to appreciate the value of a paper. The standard is that this needs to be provided by the author, not by the reader. This was a major challenge for me for many years despite working at a top research institute where I have access to very knowledgeable people. For a newcomer this is even more challenging, since you won't have anything close to the broad overview which comes from many years of experience.

I would still encourage you to try. If you're lucky you could get an editor or referee who writes some useful feedback. And if you get it published, even on arXiv, you can be extremely proud. Just don't be discouraged if it doesn't work out.

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I was in your same position in high-school, and not to come off as discouraging, but the results you have most likely do not settle the “famous” open problem you’re interested in.

In high-school, I believed I had proved P=NP by coming up with an algorithm that solved an NP-Hard problem. I even typed the solution and emailed it to a few Professors who, rightfully so, did not reply me. I had a look at that paper a few weeks ago and I chuckled at what I had wrote then. I did not understand what P, NP, correctness proofs, or even polynomial time algorithm meant, I just wholeheartedly believed I had solved P=NP.

You mention that your area is in algebra. If you have no formal training in math, the techniques you are familiar with from high-school will most likely be elementary algebra. It is likely that extremely brilliant mathematicians tried to tackle the problem, and failed to do so. Do you think they lacked your intuition or algebraic techniques?

If I were you, I would hold on to this paper for a few years and I would continue to study math. I hope I don’t come off as discouraging, it is just that peer review can be absolutely soul crushing for new researchers, and I would hate for that to happen for you when you’re still exploring Math.

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  • not 'open', I have posted the result on MSE, see the latest question about Faulbaher's formula. – Buraian Nov 23 '20 at 18:42
  • And a problem doesn't need to be "open" to be interesting. – Buffy Nov 23 '20 at 19:37
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    @Buffy I never claimed the problem must be open to be interesting? – Coconut Nov 24 '20 at 7:29
  • @Coconut The question seems clear enough: "a somewhat novel method of proving a famous theorem" (emphasis added) It's not an "open problem". The OP has updated the question to state that their solution has been verified. Also, as stated in the comment above, the OP posted the result on MSE and asked for feedback. – Brian Drake Nov 25 '20 at 11:42
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Congratulations! Even if your paper contains errors, the mere experience of writing it and submitting it for evaluation by others is hugely valuable and shows superb skill and dedication from you!

Please don't stop! I'm not clear if this is something you have already written, or is still unfinished. I am not a published researcher, but I have many friends who are published academic researchers. What I have picked up from them is that papers are never 'finished' and never 'perfect'. So please, don't aim for 'perfection' otherwise you risk never completing it.

Do a rough draft from top to bottom including the end. Don't worry about formatting or grammar. This is the creative part, where you are getting your thoughts on paper and explaining how you got from A to B. Ignore any small errors.

Importantly, wait till after you have a complete rough draft to go back and polish it. Then read it again to catch out any embarrassing mistakes or formatting issues. Then ask your grad friends or a teacher you know to read it for an outside perspective. I apologise for the repetition if you've already done all this, I'm just trying to cover the bases.

As for publishing, go ahead and publish it! Another poster, @Buffy, has mentioned the American Mathematical Society and others. If you're not American, maybe your country has a similar organisation.

Another place to publish is arXiv.org, an open academic publishing forum. They may or may not require you to get endorsement from another academic before allowing you to register with them to publish your paper. Using your school or college email address will help. If they need an endorsement, look for who is publishing papers most similar to yours on arXiv, and email 1 or 2 per day with a copy of your paper and a polite request to ask them to endorse you. Be sure to briefly mention your age and background.

Good luck! Getting a paper published on arXiv is a major achievement! It's not a peer review forum, but it's a big milestone.

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Even if the result is not new, if your proof is short (and, by definition, does not require more than what a high-schooler knows), then it might be possible to publish in say American Math. Monthly, or similar. They accept short, nice proofs.

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The OP stated in a comment on another answer:

I have posted the result on MSE, see the latest question about Faulbaher's formula.

It looks like the question they are referring to is: Faulhaber formula from geometric series and operators?

As far as I know, academic journals generally do not accept work that has previously been published elsewhere. According to a Physics Stack Exchange meta post by Emilio Pisanty:

Of course, pre-publication on Stack Exchange could make some journals and publishers refuse to publish the material, but that is a question that we cannot answer for you.

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    damn this was a true catch-22 situation. I had not many friends to discuss this with so I had posted on MSE for checking, but posting on MSE reduces my chances for getting published. – Buraian Nov 25 '20 at 13:28

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