Fascinating! This is exceptionally well-written. I can't imagine it being very easy to communicate mathematical concepts and the excitement of the discovery (!) in ways that even the uninitiated (like myself) can relate to.
His discovery (albeit not identified as such) contributed to laying the foundation for work in string theory and he had no clue.
If you liked this post, check the BBC podcast Series "A Brief History of Mathematics"[1] and also "The Music of the Primes" from Amazon. It's from the same author.
I recently bought Leonard Susskind's "The Theoretical Minimum" [0] so I could better understand theoretical physics. The explanations are short and clear, but I'm starting to realize that I learn a lot better when I know the historical context in which these concepts developed (or why they were needed). Feynman does this in his lectures and it really helps.
I agree, except the bit about string theory. I don't understand why science journalists think 'additional dimensions which are rolled up into a tiny space' means anything to the lay reader.
ics@kafka-mbp:~ ls -l ~/Projects | grep Ramanujan
drwxr-xr-x 3 ics staff 102 Nov 6 16:54 RamanujanManuscripts
Funny you should mention that... I haven't really dug in yet but I was looking for a good reason to brush up on LaTeX and have been on a math reading tear lately. I couldn't find any other attempts so if I make any progress I'll edit or comment on this with a link.
...this is exactly what Ramanujan came up with. His work on the K3 surface he discovered provided Ono and Trebat-Leder with a method to produce, not just one, but infinitely many elliptic curves requiring two or three solutions to generate all other solutions. It's not the first method that has been found, but it required no effort. "We tied the world record on the problem [of finding such elliptic curves], but we didn't have to do any heavy lifting," says Ono. "We did next to nothing, expect recognise what Ramanujan did."
There's a joke in Dr Who where the 11th Doctor (Matt Smith) gives a proof of Fermat's Last Theorem to geniuses on earth in order to convince them he can prevent annihilation by the Atraxi. He basically gives them "Fermat’s Theorem. The proof. And I mean the real one. Never been seen before. Poor old Fermat got killed in a duel before he could write it down. My fault. I slept in."
That always makes me wonder - was there and is there indeed a better, shorter and more elegant proof than the one we have?
Who knows, maybe Ramanujan was on his way to developing one!
> "He was a whiz with formulas and I think [his aim was] to construct those near counter-examples to Fermat's last theorem. So he developed a theory to find these near misses, without recognizing that the machine he was building, those formulas that he was writing down, would be useful for anyone, ever, in the future."
Hardy later described his collaboration with Ramanujan as "the one romantic incident in my life".
OMG, that is ridiculously romantic, and the linked bio of Hardy is amazing. Please tell me there is a book or play or movie about Hardy, maybe along the lines of the one Tom Stoppard wrote about Hardy's contemporary A E Housman.
They certainly were. Bruce Berndt is one example of a mathematician who has spent an entire career proving the results found in Ramanujan's notebooks. He has written a number of very thick books showing methods of proof for all his results.
Notice that Ono says that only a few people knew about this stuff. Another way of saying the same thing (the more usual way in my opinion) is that "this was already known by the experts".
They have been. The issue, as I understand it, is that when Ramanujan was doing mathematics paper was relatively expensive so he would do all his working on a slate, get to the end of the slate, flip it over, do more working, then flip over the slate and wipe off the working. When he came to his conclusion he wrote it in his notebook, with no working!
I believe in this regard he was unlike Euler, who used to go down wrong paths but explain why he was doing something and why he backed out to go down the right path (apparently Gauss looked down on this, but Gauss wasn't blind and doing it all in his head).
I think they have been. My understanding is they have been inferring the thought processes that must have laid behind the formulas written in his notebook, to find the more general theory. So they had to put quite a bit of work into this themselves.
There is a new biopic about Ramanujan. I wonder if anyone on HN knows when it will be released in the US. So far it seems it is only at film festivals.
I was very happy to see it today at a science film festival in my hometown. It's a great movie. Very dramatical. But there seems to be little info known about its release.
What do the power series at top have to do with the equations at bottom? McIlRoy of Unix pipe fame has a wonderful exposition on programming with power series. http://www.cs.dartmouth.edu/~doug/powser.html
The implied other side to the 1729 example, x^3 + y^3 = z^3 - 1 could also be thought of as a special case w^3 + x^3 + y^3 = z^3 (well, they both can, if the variables are integers). Since the more specific version has infinite examples, so must this. But what about if we bump those exponents up to 4? Is there a generalized Fermat's Last?
In fact every integer k is the sum of nine cubes, and every multiple of 6 is the sum of four cubes. Your equation is just the equation x^3 + y^3 + (-z)^3 + w^3 = 6k where k = 0.
The more general problem is known as Waring's problem. For example, every integer is the sum of 19 fourth powers. Of course special families of integers require fewer fourth powers.
Could this be further generalized to n integers raised to nth power added together can have a number raised to nth power integer, hence why there is no relationship such that x^3 + y^3 = z^3, but there are an infinite number of x^3 + y^3 + z^3 = a^3 (where z in this case can work as +-1).
a month ago(o) i mused on the potential applications of understanding why ramanujan was examining 'the smallest number expressible as a sum of two cubes in two different ways'
What a wonderful Saturday read. Thank you for posting.
The article does make a very interesting point I had never considered: Mathematics as machines. Which absolutely makes sense when you see it from the point of view of input-processing-output.
Yeah, I meant funding, sorry for the spelling. Precisely, I meant that I am afraid Ramanujan's ideas would seem so crazy that nobody would fund his research.
His discovery (albeit not identified as such) contributed to laying the foundation for work in string theory and he had no clue.