Advice for getting a paper published as a highschooler

I'd recommend not trying to submit a paper to a journal without guidance from a more experienced academic. That academic doesn't have to be a professor; anyone with more experience with the field than you should be good.

You've contacted the professor, which is a great start. If they don't reply, you can/should also talk to your teachers. Since you're fresh out of high school, there's a good chance your somewhat novel method might be wrong, or not novel. Your teachers have been in the field longer than you, so they probably know more than you; furthermore they know you personally so they'd be more likely to look at your method. If your teachers are unable to help, you could also ask friends/family for anyone with more formal training in the field.

Anyone can publish a paper, regardless of age or affiliation, provided that it meets the (rather high) standards of a journal. The standards will include things like understandable writing, but more important is whether the paper solves an "interesting" question in a "novel" way.

"Interesting" can mean new and important, or classic, or other things. "Novel" means that the approach is new, and for mathematics at least, something that might be exploited for solving other problems. A "famous theorem" is itself interesting, of course.

You don't need a professor's help to do this, but they might be able to give you some advice on your paper and how and where to present it.

But even your secondary school math teacher can probably help you with this, provided that they read enough of the literature.

But the way to get started with a publication is to submit it (probably online) to a suitable journal. You will hear from them fairly quickly if it is rejected. If they find it "interesting and novel" it will be assigned to some reviewers for deeper analysis.

A particularly challenging aspect of writing a paper is the introduction. At least in my field of research, I expect this to contain a reasonable overview of related work. Context ist very important to appreciate the value of a paper. The standard is that this needs to be provided by the author, not by the reader. This was a major challenge for me for many years despite working at a top research institute where I have access to very knowledgeable people. For a newcomer this is even more challenging, since you won't have anything close to the broad overview which comes from many years of experience.

I would still encourage you to try. If you're lucky you could get an editor or referee who writes some useful feedback. And if you get it published, even on arXiv, you can be extremely proud. Just don't be discouraged if it doesn't work out.