Try reading a few proofs. If you can put a proof on a piece of paper that isn’t simply laughed at you are in the running.

Like what is the relationship between the speed of light, and the mass of an electron. Must there not be some correlation between the potential of mass and non mass particles?

If someone could pencil that out it would be both “no duh!” And a question only an amateur would explore (of course not silly, if there was a relationship we would already know all about it.)

Maybe someone already has that one, I don’t know, not my expert area. Sure that counts, however the pursuit of Truth (or novel truths) is not played only by professionals.

https://www.smithsonianmag.com/smart-news/two-high-schoolers...

get a syllibus for degree courses, and get the c.v. of faculty, as well as theses of grad student t.a. corpus. know these things inside and out, as well as recent extensions of these theses, and when you can stand up to an examination board to make successful defense vs challange, you may be ready.

There are others like him.

If you're sufficiently smart/motivated and have a day job that's conducive to side-project intellectual effort, sure it's possible for an amateur to make contributions.

Logically? yes.

Realistically? Most likely not because scientific fields are shared human endeavors, share human endeavors are undertaken by communities, and for better or worse you are not a member of those communities.

So you are not connected to the pulse of problems people within the communities are currently interested in and don’t have relationships that will get your papers read by members of the community. Only freshmen get the benefit of the doubt when it comes to endless September.

On the other hand, publishing your work is as simple as wordpress.com or Youtube. So if the work is important to you, there are no meaningful impediments to putting it out there. People might not care, but if the work is meaningful to you that won’t matter.

Of course, the simplest way to join the community is to do the things the community expects of its members…grad school. Good luck.

https://en.wikipedia.org/wiki/Hua_Luogeng

https://en.wikipedia.org/wiki/AKS_primality_test

https://en.wikipedia.org/wiki/Richard_M._Friedberg

https://www.quantamagazine.org/decades-old-graph-problem-yie...

there's a great interview here:

https://corecursive.com/data-compression-yann-collet/#pickin...

Part of what helped Yann was finding a niche community of people with the same interest, developing new compression algorithms, where he could be part of that community:

> I was not alone and that’s very important I think. But more importantly, I think it gave me a frame of reference. I could compare, I could get evaluated. And so there was a sense of belonging to a tribe of peers and I think it matters because it’s difficult to sustain such a long effort, multi years effort with no such contacts at all.

> Every once in a while someone would come and say, “Hey, I invented this.” And some people would test it and would say, oh, it’s good or it’s great or it’s not. And there would be no shortage of people who like me are interested in data compression and would test the program. Oh, that’s a perfect ground.

Part of Yann's success in contributing something novel was having a different goal to most people

> I think it’s fair to say that in the data compression community, most searcher were interested in data compression, really best ratio. And speed, yes, as a side effect. Let’s make it not too bad. That really comes second. And my mind was refocused on I want great speed and without sacrificing speed, I want to get better compression ratio. So it really changed the perspective of what matters. And it doesn’t take too long before I got something competitive and at some point also by chance I have an algorithm which seems to be the fastest around. So that’s LZ4.

I think there's a lot of overlap with brudgers comment about fields being shared human endeavors, undertaken by communities. You need to figure out how to be part of the relevant community. For a field like theoretical physics, the cost of admission is probably grad school in theoretical physics.