Questions about Mathematicians
Let me try to give you some perspective on mathematics and mathematicians.
First, there are lots of kinds of mathematics. Algebra and Geometry were studied early in human history and other fields split off from them. At first glance these two seem quite different, but now there is Algebraic Geometry, a field of its own. There is also Analysis which studies certain kinds of relationships and Abstract Algebra which studies certain kinds of operations on things. No, there is more to it than that, but it gives you an idea. Topology is something like Analysis in some ways and something like Geometry in others. So there are some overlaps.
But the differences between mathematical fields are big enough that it is possible, even likely, that you can have true insight into only one or two of the many fields. And it is insight that is required for mathematics.
Mathematicians study problems. Applied mathematicians study &quot;real world&quot; problems and apply mathematics to their solution. Pure mathematicians study problems in mathematics itself. They study what has been shown to be true over time, but try to imagine what might be true and then set out to explore whether it is true or not. This is where insight is required.
It is unlikely, but not impossible, that you have discovered something new. It is less likely, but still not impossible that what you have discovered is significant. It may have been noticed in the past and not explored deeply, for example, seeming obvious to someone who has studied it deeply.
But, one thing that mathematicians do a lot of is work with others and share ideas. They bounce &quot;new&quot; ideas off of colleagues and get feedback. So, for someone in your situation, teachers and professors are the people most likely to be helpful at this stage.
If you want to be a mathematics researcher then put yourself on a path to eventually earning a PhD from a good school. Most pure mathematicians are also academics, though there are exceptions. Even an amateur can be a researcher, but will probably need to build up a circle of contacts and collaborators over time in order to be effective.
In a certain sense mathematicians work literally all the time. There are no breaks. But you don't sit at a desk for most of it. Your mind will work whether it feels like &quot;work&quot; at all. I often go to sleep thinking about some sticky problem and wake up with the solution. The insight comes sometimes when you let your brain relax, rather than trying to force it through some complex logical argument. The crux just falls into place.
Working with professors, especially on a doctorate, will give you a sense about what problems are important as well as ideas about how to attack them. But it is the &quot;imagining what might be true&quot; that makes a mathematician. The proofs are hard work and require training and practice, but it is that insight that makes the difference.
Only a few of us are &quot;exceptional&quot; as the word implies, or spectacular. But we all work toward excellence. Do that and you should be fine.

As a young person, I suggest you study widely (and not just mathematics). There is plenty of time to specialize later. And if you want to do serious work in any field, get in the habit of carrying a notebook (or equivalent) so that you can quickly jot down any insights or ideas that pop into your head while you are doing other things.