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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.