IMHO the likelihood that someone completely outside of any field or discipline cohort (academic/industry/etc), without existing contacts, making any kind of monumental discovery, is so vanishingly rare as to be non-existent.
So, one would already have some contact with someone of note in the field to guide disseminating the information (or to steal the idea for themselves and disseminate).
Start by going to someone in the appropriate department of your local university. There’s a high probability that your discovery is flawed, or long-known, or both, and they’ll likely be able to tell that.
In the unlikely (but possible!) event that you do have something actually new, then they can help you clean it up and get it published in an appropriate journal.
And yes, it does happen occasionally. IIRC, one of the key results in the theory of tesselations came from a stay-at-home mom with no particular math training, who liked to stare at the tile patterns on her floor and think about them.
One field where amateurs still make significant contributions to the advancement of science is astronomy. Comets, for instance, are sometimes first spotted by hobby astronomers (and then get named after them); an example is Hyakutake of 1996. The discoverer, a Japanese hobbyist, reported it to a national observatory.
I know you’re being facetious, but I think it’s safe to say that anyone who gets a Nobel Prize or Fields Medal does so as a result of a lot of hard work, and not just stumbling across something.
Well, I don’t know about any of the big awards in my case I took my idea and commercialized it and build a company around it. About 5 years later I was talking to the head of a major research institute in my field and he offered to help me publish my idea in a major journal and we’re working on the article right now.
That was Marjorie Rice. She learned about tesselations from reading Martin Gardner’s Mathematical Games column in Scientific American and sent him her results. He passed them on to experts and wrote about her discoveries in one of his books.
Note that tesselations are one of the extremely rare items in math that can be done without any “real” math. The experts could barely follow her work because she didn’t know the standard notation. Rice could easily have been dismissed because of that.
There’s nobody like Gardner these days, somebody nationally famous for engaging in a massive correspondence with amateur enthusiasts. The best advice is that given by Chronos: go to the lowest level of local expert who will listen and have it work its way up.
And that’s part of what I was referring to when I mentioned that a professor could help you clean up your work for publication: They could help you put things into notation that others in the field would understand.
Brian Josephson was just a graduate student when he published his prediction of what became known as Josephson tunneling in 1962. It was disputed by multi-Nobelist John Bardeen, but confirmed experimentally in 1963. He was awarded the Nobel prize for this prediction in 1973.
It’s an astonishingly simple theoretical prediction. But at the time, everyone assumed the currents they were seeing were “super shorts”, not supercurrent tunneling.
Claude Shannon basically invented digital logic (with vacuum tubes) for his Master’s Thesis.
As for the OP, I’d read papers in the field, then try to write the new discovery up as a paper, which will involve looking for prior work, and then take it to the local university.Nobody’s getting a Nobel Prize without publication.
Ramanujan wrote to G.H. Hardy and his work (a body of work, not a single result) got him an invitation to Cambridge. There is at least one real life story (sorry, I have forgotten the name) that was like Good Will Hunting. A janitor saw a problem on a blackboard and solved it. If Fermat really had a valid proof (which I highly doubt) someone could rediscover it. The actual published argument would take me years to understand and cannot be discovered by an amateur.
Fermat did have an actual proof… for the n=3 and n=4 cases. Which he went on to publish. Most likely, when he wrote his margin note, he was assuming that the same method would generalize to all n. When he later put in the work and discovered it didn’t, he published what he had, but never bothered to retract his margin note, because after all, it was just a margin note.
The professor in the case of Majorie Rice was Doris Schattschneider. I have met her and was awarded one of her books for my achievements as a mathematics undergrad.