Discovered that my research was not so original after it was complete. Is there a policy I should follow here?

This happens even to people who know how to search the literature and have some experience. Often it happens because when one starts thinking about something new, one understands little, and only once one has understood some things is it possible to realize that others have also already studied and understood them. Talking to other researchers, attending conferences, reading widely, etc. are all activities that help avoid rediscovery, but obviously they are all hard for a high school student.

Generally, one should not publish results that are already available in the literature, particularly if one's methods are in the fundamental aspects the same as those used previously (moreover, it will usually be quite difficult to publish results that are already known). There are exceptions, as sometimes one has a new take on an old result, or a better way of obtaining it, or sometimes one can adapt one's work into an expository account of the already known results that could be published (this will work only if those results have generated a fair amount of interest). If one succeeds in publishing already known results, it could generate in others' minds doubts about one's honesty, although it certainly happens that people innocently rediscover and publish known results (I know several examples). Often this occurs when they rediscover them in a context different from that of the original discovery (e.g. a graph theorist discovers some algebraic fact, and publishes it in a graph theory journal, although some experts in the relevant algebra would recognize it as a special case of something they already know).

However, there is a positive side to such rediscovery, and this is the most important aspect for a high school student.

The positive aspect of rediscovery is the following. First, that one has rediscovered published results should confirm one's sense (that one ought to have already) that one is doing something interesting (this is important for one's confidence). Second, any interesting results admits extensions, variations, further developments, etc. so such rediscovery provides a base for further work. Third, as a practical matter, that someone else has already worked on something means the audience for extensions of those results is at least one (besides oneself) and it is always positive to reach others with one's work. So the productive/constructive response to discovering that one has rediscovered something is to push it farther. This really works in practice. My most recently accepted paper started this way - I worked out something, it seemed so natural that I thought this surely is in the literature already, and, sure enough, there it was, published 5 or 10 years earlier. It was a disappointment, but I kept thinking along the same lines, and it was possible to build on it to obtain something new. Also it brought me into contact with the authors of the previous work, which for me was also positive.

Finally, I would recommend that you get in touch with the mathematics department at the nearest research university. Usually someone is happy to help out an enthusiastic and talented student.

First of all, congratulations on continuing to look for related work. Good job finding the right search terms.

If you were going to publish or submit for a competition you would have to edit your paper to describe the prior work and reference the forum post. I suggest editing it accordingly. Now evaluate your paper to see if it meets the requirements for the target venue.

If not, do not attempt to publish. You still learned from doing the research, so don't think of it as a waste of time.

If you do decide it meets the requirements for some venue, you can submit it with a clear conscience provided you discuss and reference the prior work.

In the long term, Internet searches will only be one way you will have to find related work. When, as a graduate student, you are required to publish original research, you will have an academic advisor who has contacts and experience to help you. You will take courses, read journals, and go to conferences, during which you will learn about alternate approaches to your specialization. It is not 100% - there are plenty of cases of results being discovered and published more than once - but you will have more resources.

Mostly agreeing with Dan Fox, so only going to comment on top of his answer.

In short, my advice would be: don't try to push the paper out too hard. Do publish it, but on your website or arXiv. Be sure to emphasize that it is a rediscovery and be generous with credit to the first paper. Here are the reasons why i think this:

• If you've rediscovered something, chances are that somebody will again, especially if the first paper is indeed hard to discover. You publishing this anywhere visible, perhaps using more current terminology in the title/keywords, might save the needless work to this somebody.
• Being such a young researcher, you have very little pressure to produce journal publications for now. A few years down the line when you perhaps decide to pursue an academic career, it's hard to imagine one ranked paper will make the difference in a hiring process.
• Going fully transparent and not trying to 'milk' the work as much as possible might actually give you credit for being honest. Yet it still increases the visibility of the work, which (assuming it's useful as such) benefits both you personally and the community.

One more comment, this time more general. I believe that young researchers today focus too much on 'producing results'. And understandably so, given how overly competitive science has become. I see this trend especially among fresh PhD students, and sadly, their advisors often fuel this as it obviously benefits them too.

I'm not saying this is your case, but regardless, try to resist being pulled into the vortex too fast. Early stages of a scientist's career should mainly be about discovering new things, studying as much as possible, and developing complementary skills. Publishing top papers might often conflict with that, because it requires undivided and long-term focus.

Becoming a scientist is a noble goal, but one that shouldn't come ahead of becoming a well rounded person with developed world view, ethics and such old-fashined things. And if it seems i'm preaching too much, well perhaps so, but i'm simply saddened how many young and talented people succumb to the fashion of rushing into scientific limelight while on the inside their egos are stunted and suffer.