irchans

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Was a PhD necessary to solve outstanding math problems?

I'm not sure what you mean by "a basic formal education". Ramanujan attended Government Arts College, Kumbakonam and Pachaiyappa's College and then he became a mathematical "researcher at the University of Madras". About 3 years later, Ramanujan wrote his first letter to Hardy. It is certainly true that he did not have the normal formal background of a Ph.D. student, but I think he did take a number of undergraduate math classes.

Was a PhD necessary to solve outstanding math problems?

I started my Ph.D. at age 27. I would have made a lot more money during my career if I had instead just joined a large company like Microsoft because I was a good programmer, but I really have enjoyed mathematics.

Was a PhD necessary to solve outstanding math problems?

I think that math is a "young person's game" because it is intellectually demanding requiring a fair amount of energy to process what is going on. It takes years to learn the background necessary to do the research, so I think that most discoveries are made by mathematicians in the age range of 30 to 45.

I'm 55 and I have mostly given up on doing research. I just don't have the energy and I've never been a professor, so I just don't have the time to do it for fun anymore. My former advisor continues to publish in his 70s.

Was a PhD necessary to solve outstanding math problems?

to get a PhD is many times easier than to "accomplish groundbreaking work"

Shminux, I want to disagree with this statement. I guess it all depends on what you mean by "groundbreaking work". First, it seems to me that most of the Ph.D. dissertations that I know of entailed discovering new theorems. Second, in my math department, it took 7 years on average to get a Ph.D. It often takes a lot less time to discover and write up a good, popular paper (say one with over 20 citations).

If you don't consider a paper "groundbreaking" unless it has over 100 citations, then maybe you are correct. I just don't know.