Mathematics and some areas of physics and computer science have the peculiar appeal that some problems and results are easy to understand and it is conceivable that somebody armed with nothing but the right idea can come up with something groundbreaking. This makes these fields particularly prone to amateurs becoming obsessed with solving famous problems or debunking something established. Some of these erroneously think they succeeded and then proceed to pester professional scientists (and are typically called cranks or crackpots).
Now, there are often good arguments that it is unlikely that an amateur is actually onto something, but here I am wondering about the empirical side:
What are the most relevant results (if any) produced by amateurs in these fields? Some specifications:
Eligible results must be pen-and-paper theory, possibly aided by a computer.
Finding a simpler or completely different proof for a solved problem is eligible.
Results that can in principle be found by brute-force computing are not eligible (even if they require some search strategy to avoid combinatorial explosion). It’s not that such results are without value, but they do not fit what I am interested in for several reasons (results are easy to check; it’s more plausible to get lucky; obsession may be a virtue; …).
For the purpose of this question, an amateur is somebody who never did any of the following:
- acquire an academic degree in a field that features proofs (mathematics, physics, computer science, …),
- publish a paper in one of these fields,
- make a living of performing research in such a field.
The result must have been found after 1960. This is not a hard deadline; I mainly want to ensure that the basics of mathematics and physics had been thoroughly explored and to somewhat ensure that there were no amateurs who would not be amateurs nowadays.