High School Students Publishing Mathematics Right now I'm a senior in high school and I will be submitting a number of applications for undergraduate admissions with deadlines ranging from November $30^{\text{th}}$ to early February. However I recently had the chance to talk with a number of rising high school seniors who plan to major in physical sciences e.g NMSC finalists and other national awardees, and it seems all of them claim to have conducted "research" in there planned field of study. Whether it be shadowing a graduate student or even claims of co-authoring papers, the whole thing just seems very silly to me. I mean the idea that a high school student is capable of finding or writing anything substantial seems far-fetched in my opinion. With that said I am sure there are obviously a few special cases with students who are exceptionally "gifted", of whom have actually written something substantial. But the shear number of people I have been talking to who claim to have conducted "research" in one form of another just seems unrealistic to me.
So my problem is that i'm really worried I might not be making my applications realistically competitive with other students who are applying to many of the schools that I would like to attend. Now generally speaking I am not at all familiar with academia and much more so I am clueless when it comes to the topic of research, but I feel like I should try and write/submit something for my applications. Personally, I have been self studying mathematics for about $4$ to $5$ years now, not including general college preparatory mathematics, and through this time I have taken note of curious identities as well as interesting bits/proofs/ideas, so I want to know if there is someway I can use any of this and possibly turn some of it into a paper and then get it 'published' or something, so I can put that on my application.
So my question is how would I even start to do something like that?
Also I am sort of confident most of the stuff I have written is likely un-substantial or trivial in nature, if I did 'publish' something or 'submit' work. Would there be any way I could take it down in the future? Out of fear of possible embarrassment somewhere down along the road when I become more educated, at which time I wouldn't want a paper like that floating around. If it helps I posted several of the identities/bits bellow I have scribbled down in the past, to help anyone recommend a place where I could 'submit' or 'post' something of this nature.
$$\frac{1}{\pi}=\frac{1}{3}-8\sum_{n=1}^\infty e^{-2\pi n^2}n\coth(\pi n)-2\sum_{n=1}^\infty e^{-2\pi n^2}\text{csch}(\pi n)^2$$
$$\sum_{m=1}^\infty\frac{1}{m^7}(\frac{\sinh(\sqrt{2}\pi m)+\sin(\sqrt{2}\pi m)}{\cosh(\sqrt{2}\pi m)-\cos(\sqrt{2}\pi m)})=\frac{13\sqrt{2}\pi^7}{56700}$$
$$L(2,\chi_4)=G=\frac{11\pi^2}{120}+6\sum_{m=1}^\infty\frac{e^{2\pi m}}{m^2(e^{2\pi m}-1)^2}$$
$$\sum_{n=1}^\infty\frac{x^{2n}q^n}{y^{-2}-q^n}=\sum_{n=1}^\infty \frac{y^{2n}q^n}{x^{-2}-q^n}=\sum_{n=1}^\infty (xy)^{2n}q^{n^2}\frac{(1-xyq^n)(1+xyq^n)}{(1-x^2q^n)(1-y^2q^n)}$$
 A: The only part of your question which I feel I'm qualified to answer is the last paragraph:
Yes, whatever you write now you will likely look back on as insubstantial or trivial. However, you should not be afraid of your junior accomplishments. Nobody will hold the quality of your first papers against you - you will be judged on the basis of either your most important or your most recent work. This is actually very important, because if you are aiming to seriously go into some field - whether it is pure math or something else - you will, at some point, have to write your first paper. If you are afraid of embarrassing yourself, you will never get it done.
I know someone who published a paper in high-school. Sure, it wasn't very good, but a few years later down the line, he's really made a name for himself, and is enjoying a very prestigious position at a very prestigious school. His career is well on its way, and I'm sure he would agree that it's important to start early.
And don't forget: if you're really afraid of embarrassing yourself, remember that you can excuse yourself by referring to your relative academic immaturity - as you are still in High School. Nobody will hold your work against you.
