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.