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Question:

How can an independent researcher publish a well-written research paper / proof in a respected mathematical publication?

Related Questions:

How do I ensure I'll be able to claim authorship over a math proof, as an amateur math student? How can I "time-stamp" my data without publishing it?

I've reviewed these both, and while I find them somewhat insightful, don't believe they answer my question directly. Generally speaking, how should an independent researcher go about submitting to an academic publication?

Suggested Duplicate / Existing Answer:

I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?

Whether a "crank" could possibly write a valid proof of this nature is outside the scope of this question. Simply how and where an independent researcher should submit their work when they believe it is of sufficient quality to do so.

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    You are in a tough spot: Given what you wrote, you have hard time even writing coherent math. Most likely (with > 0.99 probability), whatever you will write, no professional mathematician will read. My suggestion would be to forget the Collatz Conjecture and try to learn some rigorous math maybe by reading a book or online notes with an introduction to math proofs (you will find many just by googling). Right now, you are in a situation akin to somebody who would like to play, say, basketball, at a professional level, but cannot dribble a ball. Sep 8, 2023 at 5:04
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    My suggestion: learn multivariable Calculus and complex analysis online first. Do a lot exercises for them. Then go back to your proof to see it makes sense or not. Right now, you don't even have the foundational knowledge to tell if your proof is correct. If you go to a professional mathematician for a co-author, it's very likely you will be laughed at. So, have enough knowledge to review your own proof before you go to anybody else.
    – Nobody
    Sep 8, 2023 at 6:08
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    No, I am not suggesting looking for a coauthor (at best, you will find someone incompetent). I am suggesting learning how to do rigorous math. Sep 8, 2023 at 12:43
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    Moishe was translating from one event to another event of similar probability but for which the vanishingly smaller probability of the latter event is easier to believe. I did this myself upon reading your question. What I came up with is: it is about as likely that you have solved the Collatz Conjecture as that in the first full tennis match of your life, you beat Novak Djokovic. Yes, it is possible, but it is overwhelmingly more likely that some mistake has been made. If you are serious about tennis, take the process of improvement seriously... Sep 9, 2023 at 5:01
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    @TCooper: Bigoted? That is a very heavy word to apply to someone who is giving academic advice to an anonymous person on the internet. I think I had better not engage further. Sep 9, 2023 at 15:48

3 Answers 3

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If a professional mathematician had a proof of the Collatz Conjecture, they would submit a paper to the Annals of Mathematics. You can look at their webpage for submission guidelines, but I believe it just involves sending a PDF of your paper to the appropriate email address. There are no costs.

I am about 99.99% certain that there is no elementary proof (one that does not involve significant techniques from real and or complex analysis) for the Collatz Conjecture. If someone submitted such a purported proof to the Annals of Mathematics, particularly if it's an amateur, they won't bother to look at it because it's almost certainly a waste of their time.

Any other reputable journal would do the same thing because it's also a waste of their time (plus if you really believed your proof had a chance, you would've submitted it to the Annals of Mathematics).

You should know that the vast majority of mathematicians will not submit a proof to a journal unless they are 99.95% certain their proof is correct. It is very unusual for a paper to be rejected only for errors in a proof (though it does happen, more than 0.05% of the time, indicating most mathematicians are overconfident about their proofs). You don't seem to have that level of certainty in your work.

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    @TCooper - I think you misunderstood my statement - for the purposes of that sentence I am defining the word "elementary" to mean not using significant techniques from real or complex analysis. Presumably, you do know whether you're using real/complex analysis or not, so you should know for certain, not just believe, whether your proof is elementary or not according to the definition stipulated for the purposes of this argument. Sep 8, 2023 at 5:13
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    If you cannot understand this paper: arxiv.org/abs/1909.03562 you did not prove the Collatz Conjecture. Sep 8, 2023 at 5:31
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    @TCooper - yes your summary of the abstract is correct. That paper does not prove the Collatz Conjecture, though I think the experts in the area believe it is significant progress towards a potential proof. Keep in mind that a professional would need at least a week or two to actually read the whole paper and understand all its details. It's relatively easy as math papers go - I actually would have some chance of understanding it despite my expertise being in a completely different area of mathematics. Sep 8, 2023 at 5:54
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    @TCooper: "I'm not sure what qualifies as significant techniques from real or complex analysis. I've done extensive research and analysis..." Sounds like you may be mistaking the branch of pure math known as "Analysis" en.wikipedia.org/wiki/Mathematical_analysis#Main_branches with everyday uses of the word. If so -- if you're not familiar with the branch of math called analysis -- it's implausible for you to have independently reinvented the tools (built over centuries of work) needed for problems like this one. Take a course like "Real Analysis" first.
    – civilstat
    Sep 8, 2023 at 18:51
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    @TCooper - sounds like you're looking at limits under the 2-adic norm - something Tao touches on briefly as a well-studied but not ultimately fruitful approach in his paper. Sep 9, 2023 at 3:33
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Direct answer to the question that's asked: you write the paper up and submit it to a journal. There's no requirement that the author be affiliated to a university to publish.

How do you submit? You go to the target journal's website and check their submission procedure. There's a high chance you'll need to register an account on the journal's editorial management system, and then there's a submission system not dissimilar to the kind of system you might use to apply for university admission or for a visa to some foreign country.

Which journal? Find something where your proof is within the aims and scope. In the case of the Collatz conjecture, I'm not a mathematician, but a proof would apparently be one of the greatest mathematical achievements of all time, which means you should send to the most prestigious mathematics journals out there - so e.g. Annals of Mathematics. Note the website gives exact submission instructions (in this case you don't need an account at all, you can just send them an email).

Finally, you might be interested: Do journals in general have any kind of policy regarding papers submitted by someone without a research affiliation? Top journals get lots of submissions. They won't desk reject a submission solely because you are an independent researcher, but they will desk reject if your paper is poorly written, poorly formatted, poorly structured, etc. When you say your paper is well-written, sure it really is well-written (not just English-wise, but also in the "language of mathematics"), or the reader might not make it past the abstract.

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Try to follow the rules for writing using available papers from the field as an example and follow the rules from guidelines for authors from the relevant journals. Then submit your work to preprints.com or arxiv.org.

Don't worry after that. You will find a way to real, reviewed journals if the work is correct and a valid extension of knowledge in the field.

One shouldn't undermine themselves with such statements as "I'm not real mathematician". If you reached the proof as you said, then you are. Initially many "real" scientists will not understand your findings and will tell you that it is nothing more than rubbish if you haven't yet provided any proof.

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  • Hate to see all the flack you're getting for a valid answer. It's odd to me how often hard problems are solved by people outside the field offering a fresh perspective, and yet people within the field will continually assume only they themselves, with the tools that have failed them for decades or centuries to arrive at a solution, are the only possibility. Sure, in most cases building off existing work is the only way forward - but for the hardest problems, it's most often a novel approach from a novice that, at least starts the ball rolling, if not providing a direct solution.
    – TCooper
    Sep 10, 2023 at 20:34
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    @TCooper if you know of examples of a hard problem being solved by someone outside the field offering a fresh perspective, you should write an answer to this question: academia.stackexchange.com/questions/181336/…
    – Allure
    Sep 10, 2023 at 23:10
  • @Allure Rather than citing examples that are anecdotal, here's scientific research covering numerous examples -journals.sagepub.com/doi/full/10.1177/0956797616634665 - You can also check out InnoCentive - en.wikipedia.org/wiki/InnoCentive - who collaborated on this research. If a counter-argument is simply "this doesn't apply to complex mathematical problems because math is special", I have no contradiction for a logical fallacy.
    – TCooper
    Sep 10, 2023 at 23:30
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    @TCooper again, if you know of examples, you should write them as answers to that question.
    – Allure
    Sep 10, 2023 at 23:38
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    @Allure I guess I missed the point of your comments in this context originally. It seemed you were attempting to discredit my statement by implying a thread on this site had no specific examples meeting the criteria. I was simply attempting to support my statement. I have no interest in copy and pasting 230 examples from the research I linked to that thread, or talking about the petroleum engineer that created auto-tune, etc. The fact that outsider's fresh perspectives often lead to novel solutions is well established.
    – TCooper
    Sep 10, 2023 at 23:52

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