I think the “sudden and inexplicable flashes of insight” description of Ramanujan is exaggerated/misleading.
On the first example of the post: It’s not hard to see that the problem is, by the formula for triangular numbers, roughly(!) about the solvability of
x(x+1)2=n(n+1)4.
Since t(t+1) is roughly a square - 4t(t+1)=(2t+1)2−1 - one can see that this reduces to something like Pell’s equationa2−2b2=1. (And if you actually do the calculations while being careful about house x, you indeed reduce to a2−8b2=1.)
I think it’s totally reasonable to expect an experienced mathematician to (at a high level) see the reduction to Pell’s equation in 60 seconds, and from that making the (famous, standard) association to continued fractions takes 0.2 seconds, so the claim “The minute I heard the problem, I knew that the answer was a continued fraction” is entirely reasonable. Ramanujan surely could notice a Pell’s equation in his sleep (literally!), and continued fractions are a major theme in his work. If you spend hundreds-if-not-thousands of hours on a particular domain of math, you start to see connections like this very quickly.
About “visions of scrolls of complex mathematical content unfolding before his eyes”: Reading the relevant passage in The man who knew infinity, there is no claim there about this content being novel or correct or the source of Ramanujan’s insights.
On the famous taxicab number 1729, Ramanujan apparently didn’t come up with this on the spot, but had thought about this earlier (emphasis mine):
Berndt is the only person who has proved each of the 3,542 theorems [in Ramanujan’s pre-Cambridge notebooks]. He is convinced that nothing “came to” Ramanujan but every step was thought or worked out and could in all probability be found in the notebooks. Berndt recalls Ramanujan’s well-known interaction with G.H. Hardy. Visiting Ramanujan in a Cambridge hospital where he was being treated for tuberculosis, Hardy said: “I rode here today in a taxicab whose number was 1729. This is a dull number.” Ramanujan replied: “No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.” Berndt believes that this was no flash of insight, as is commonly thought.He says that Ramanujan had recorded this result in one of his notebooks before he came to Cambridge. He says that this instance demonstrated Ramanujan’s love for numbers and their properties.
This is not say Ramanujan wasn’t a brilliant mathematician—clearly he was! Rather, I’d say that one shouldn’t picture Ramanujan’s thought processes as wholly different from those of other brilliant mathematicians; if you can imitate modern Field’s medalists, then you should be able to imitate Ramanujan.
I haven’t read much about Ramanujan; these are what I picked up, after seeing the post yesterday, by thinking about the anecdotes and looking to the references a little.
I think the “sudden and inexplicable flashes of insight” description of Ramanujan is exaggerated/misleading.
On the first example of the post: It’s not hard to see that the problem is, by the formula for triangular numbers, roughly(!) about the solvability of
x(x+1)2=n(n+1)4.
Since t(t+1) is roughly a square - 4t(t+1)=(2t+1)2−1 - one can see that this reduces to something like Pell’s equation a2−2b2=1. (And if you actually do the calculations while being careful about house x, you indeed reduce to a2−8b2=1.)
I think it’s totally reasonable to expect an experienced mathematician to (at a high level) see the reduction to Pell’s equation in 60 seconds, and from that making the (famous, standard) association to continued fractions takes 0.2 seconds, so the claim “The minute I heard the problem, I knew that the answer was a continued fraction” is entirely reasonable. Ramanujan surely could notice a Pell’s equation in his sleep (literally!), and continued fractions are a major theme in his work. If you spend hundreds-if-not-thousands of hours on a particular domain of math, you start to see connections like this very quickly.
About “visions of scrolls of complex mathematical content unfolding before his eyes”: Reading the relevant passage in The man who knew infinity, there is no claim there about this content being novel or correct or the source of Ramanujan’s insights.
On the famous taxicab number 1729, Ramanujan apparently didn’t come up with this on the spot, but had thought about this earlier (emphasis mine):
This is not say Ramanujan wasn’t a brilliant mathematician—clearly he was! Rather, I’d say that one shouldn’t picture Ramanujan’s thought processes as wholly different from those of other brilliant mathematicians; if you can imitate modern Field’s medalists, then you should be able to imitate Ramanujan.
I haven’t read much about Ramanujan; these are what I picked up, after seeing the post yesterday, by thinking about the anecdotes and looking to the references a little.