How are contributions to mathematics reviewed/published? Let's say you're a hobbyist who's seemingly found a novel identity/solution/method or other general contribution to some field of mathematics. What you don't know however is:

*

*If you're actually the first to discover it

*If it is useful

*If it isn't useful, is it still worth having reviewed?

*How to have it reviewed

*How protective should you be of your finding?

*If it is useful, how do you ensure you're accredited for it?

*If, how and when you should post it on stack exchange

While I am asking for generalized advice I'll offer some background for context: I'm an engineering student, and I enjoy trying to find solutions to non-elementary integrals via functional analysis, complex analysis, combinatorial methods, etc. simply because I like the challenge and I've learned some fascinating things about other fields of mathematics I wouldn't normally be exposed to. With that being said, I may have managed to find a solution to a couple of non-elementary integrals and while I'm aware they have limited application, I'd still like to know if my work is correct and worth sharing, and maybe even get credit if it would land me a scholarship or something. The last part may be simple wishful thinking, but who knows?
 A: Many of these things will require advice or scrutiny from a mathematician with experience in the field.  Since you are a university student, you should have easy access to a number of academics in the mathematics faculty, who regularly publish papers on mathematics.  I recommend speaking to one of them directly about your idea, but in the meantime, here is some general advice.

What you don't know however is:

*

*If you're actually the first to discover it.


That is something that we would typically find out by doing a "literature review", where we look at other papers/textbooks in the topic and see if we can find the result.  For this part, seeking review by an experienced mathematician in the field will probably be helpful, since they may recognise the result, and if they don't, that goes some way towards suggesting that it might be new.
If you perform a literature review on the matter, one of two things will happen.  Either you will find that the result is already published, or you will not find it.  In the latter case you would typically seek publication of your result, cite relevant literature in the general field, and report that you were unable to find the result in any of the existing literature.  Reviewers for the journal will then scrutinise this, and they will be a second check on whether it is an original result.  (And if you are like I was in grad-school, you will get a nice polite desk rejection from a helpful journal editor alerting you to the non-originality of your work!)



*If it is useful.


That is something that requires professional judgment, but if you can point out an applied use for the result (where it performs better than other approaches), then obviously that shows that it is useful.  The best thing to do here would be to see if you can form an applied example of your result, showing that it can be used to solve a practical problem.
On this point, it is also worth noting that papers on non-original mathematical results can still be useful if they provide an alternative proof of the result that is new.  Papers can also be useful as educational publications even if neither the results nor their proofs are original, so long as they simplify a problem, or present it more clearly than existing works, or provide some useful insight that improves understanding.



*If it isn't useful, is it still worth having reviewed?


Maybe, maybe not.  There is a history of mathematical results being published that were not perceived at the time as having any useful applications, but later came to have useful applications in unexpected ways.  This was particularly so in early number theory.  In order to form a view on whether "non-useful" mathematics is worth having reviewed, you might wish to read G.H Hardy's famous work, A Mathematician's Apology.



*How to have it reviewed.


Before getting to this point, you will need to write up your work in a professional mathematics paper written in an appropriate form.  This would usually entail some context and literature review, and then clear presentation of your results and proof, and discussion/examples of applications.  Once you get to this point you can submit your work to an academic journal appropriate to the paper.  If you are not sure where to submit, ask a mathematician at your university, look at where other papers on the topic have been published, and browse some articles in possible target journals.
Once you have written your paper and identified an appropriate journal, read their guidelines for submission and go through their submission process.  Typically your paper will first go to an editor for an initial "desk review" and if they like it then they will send it out to independent referees for a full review.  Rejections in this process are common (even if your paper is good) and so ---assuming your work is publishable--- you might need to make revisions and try a few journals.



*How protective should you be of your finding?


You can't simultaneously publish your findings and also be protective of them, so you will have to make a choice of what you want to do.  If you are particularly concerned about having your result "stolen" then you can publish the basics of it (e.g., theorem and proof) on a website or arXiv paper to establish precedence, but it is probably premature for this.  It would be best to at least find out if it is a new result first.



*If it is useful, how do you ensure you're accredited for it?


If the result is original and useful then it should be publishable, and the authorship of your work will automatically give you credit for the work.  Again, if you are particularly worried about having your result "stolen", you can publish the basics of it (e.g., theorem and proof) on a website or arXiv paper to establish precedence.



*If, how and when you should post it on stack exchange.


This depends on whether or not your work is amenable to a simple Q&A appropriate for this site.  Some results are sufficiently simple that they can serve as a useful Q&A  post, but many research results are more complicated (or voluminous) than is appropriate for this site.  In any case, posting your result as a Q&A on this site would serve the function of establishing initial publication, and subjecting your work to critical scrutiny by experts.  If you think this is a good idea then presumably an appropriate time to do so would be when you have a clearly formulated question, and an answer consisting of a theorem and proof.  It is possible, but I think unlikely, that this might undermine publication in a journal.
A: Write it, submit it, and find out!
I'm in a similar situation - I am an amateur mathematician and I had an idea I wanted published, and was successful!  My paper was "Extending the Algebraic Manipulability of Differentials" and it was published in "Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis".
Here's the simple answer - write it up, submit it, and find out!  If it is rejected, they usually say why.  If you are part of an academic organization (i.e., a student), then I would find a professor who would help you initially, and then find the right journal.  Otherwise, I would write it up.
Another thing you can do is find a mathematician on UpWork.  For my own paper that's what I did.  Trying to get someone to give you worthwhile feedback is hard.  If you find a professional mathematician who is willing to look at it, chances are they are just going to be critical, because you probably didn't word things the right way, and mathematicians tend to be incredibly pedantic.  Paying someone is helpful because then they are trying to make you successful, and will help you fix it up and word it in a way that makes sense, rather than just telling you that you are "wrong" just because you don't have the precision in terminology that a professional would.
Also note - there are many good ideas which, while not new, certainly could use re-explaining or just reminding everyone of them.  You can often get these published as education papers.  That is, you aren't claiming that it is new, you are talking about math education and how it can be improved by focusing on this idea, or that the idea can be better explained to students by going a different route than usual.
