How did ramanujan die?

Answer 1

I've read a different version of the 1729 story, in which someone was visiting him and they remarked on the unremarkable quality of the number, which Ramanujan contradicted with the interpretation you give [roughly]. There's certainly a chance he had played with such numbers earlier and remembered previous reasoning, rather than coming up with it on the spot. Even the story I read is a bit apocryphal; it should really have a reliable citation.

I also remember reading a slightly different version of your house number problem, where Ramanujan says that he immediately knew the solution must be a continued fraction, and solved the whole class of problems. That version didn't actually describe the problem, and I have trouble following your description; sorry.

It's probable that Ramanujan only wrote his conclusions in his notebooks, having worked the results out elsewhere. I'm interested in the 5 formulas you say haven't been proven: could you list them?

I find that not having access to a computer forces ingenuity in performing calculations. For instance, in the story about Gauss summing the numbers from 1 to 100, if he had access to a scripting language he might have just had a computer find the answer instead of figuring out the triangular number formula. In any case, perhaps the lack of access to computers helped Ramanujan in some way.

I don't believe your assertion that he never showed any proof of his work. I wish I had access to some of his published work to see for myself.

Ramanujan was very rare and had an amazing intuition. Comparing him to current university math students is like comparing them to Gauss--it's an unfair comparison.

How would math be different if he had survived for longer? It's completely unclear. He probably would have continued making numerous deep and surprising discoveries related to analytic number theory. Perhaps something would have been groundbreaking, or perhaps they would just be interesting curiosities. Who can say?