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I am an amateur mathematician, not a member of academia. I've created a small algorithm for a twin prime sieve. It's like a prime number sieve, but for twin primes of the form (6k-1, 6k+1). However, those details aren't pertinent to this question.

I wrote a paper using LaTex and a template for a math publication, naively submitted it for consideration and was rejected with no reasons given.

With no basis for modifications, no reason to believe submitting to any other publication would be successful, and because publishing is not really a serious driver for me, I naively decided to throw it on Wikipedia.

The reviewer for Wikipedia made a comment that the information might be notable, but without citations to articles discussing it, there was no way to tell and the article could not thus be published on Wikipedia.

As I said, formal publication isn't that important to me. However, my experience leaves me wondering how a person in this situation actually could publish. No publication -> no citations -> no publication allowed. It's a catch-22 situation.

At this point, I'm more interested in publishing as an exercise in figuring out how to actually do it. What if I thought up something that was actually useful? How would I go about it?

One challenge is, not being in academia I have no research resources other than Google or a public library. As far as I can tell my algorithm is original.

Second challenge is, being original, I have no sources to cite in a bibliography.

Third challenge, I suppose, is lack of a more senior academic to serve as my advisor.

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    The "catch-22" you describe seems to be specific to your attempt to "publish" on Wikipedia. And that's simply because Wikipedia is not, and was never intended to be, a place to publish original research. There is nothing stopping you from publicizing your work on your own blog, or on viXra, or anyplace else whose policies permit posting such material. Nor do academic journals have any requirement that your work be "cited"; indeed they will specifically refuse to publish anything that has already appeared in another journal.
    – Nate Eldredge
    Jun 1, 2017 at 3:20
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    Nate Eldredge's comment is very good, and should be an answer. About "challenge #2:" I don't understand this -- so much has been done on prime numbers that you should have quite a lot to cite on various approaches people have taken to sieves, and how yours is similar or different. About #1, the general public can usually make use of university libraries, though it may necessitate going in person and filling out forms. Plus, there's lots of books on number theory out there!
    – Raghu Parthasarathy
    Jun 1, 2017 at 3:53
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    Did you try posting to arXiv?
    – JeffE
    Jun 1, 2017 at 11:15

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You might want to try and win over a co-author. An expert in the field can provide the needed background, the literature review you might be missing and might also be able to tell you if your paper is really worth publishing. Maybe what you wrote is clear for every expert on the field, due to some results by someone that you simply don't know of?

Small advice here: People who are working on a PhD or just got one recently have often more time and are more willing to read something that might potentially lead to a publication than (old) professors.

Next, you might try to contact the journal you send your article to. Maybe they are able to explain the rejection?

Last but not least: If your idea is short enough to put it on Wikipedia, you might try to post it (or at least part of it) on Mathoverflow. If you follow the rules there, you might find someone who is able to tell you if this is really something "new" or if it is obvious/wrong/...

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