Astronomy and mathematics are two of the remaining professional academic disciplines that are still open to significant contributions from amateurs, hobbyists, and other unaffiliated individuals. This is likely to do with the nature of research in those fields, such as the fact that almost everyone has access to the sky, or the ability to experiment with abstract mathematical ideas. It is not just about things professionals have not 'bothered to check', but rather there is still so much to learn out there in astronomy and mathematics that people from a wide variety vocations can "stumble" across new discoveries. See this list
of amateur mathematicians and their original/primary vocations. You have primary school teachers, lawyers, an innkeeper, as well as engineers and scientists from other fields and many many more.
I encourage everyone who has an interest in astronomy and/or mathematics to dabble in those areas with the understanding that they too can still make new discoveries. Try joining your local astronomy club, or an amateur mathematics society
, or just keep dabbling and observing, and don't think "I could never make a real
discovery," because it is