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A Last Gift from Ramanujan

Srinivasa Ramanujan is a legend of the mathematics world. The son of a shop clerk in rural India, he taught himself mathematics, primarily out of a book he borrowed from the library. The math that he did started out as rediscovering old results, and then became novel, and ultimately became revolutionary; he is considered to be one of the great minds of mathematical history, someone routinely mentioned in comparison with names like Gauss, Euler, or Einstein.

Ramanujan's work became known beyond his village starting in 1913, when he sent a letter to the British mathematician G. H. Hardy. Ramanujan had been spamming mathematicians with his ideas for a few years, but his early writing in particular tended to be rather impenetrable, of the sort that today I would describe as "proof by proctological extraction:" he would present a result which was definitely true, and you could check that it was true, but it was completely incomprehensible how he got it. But by the time he wrote to Hardy, both his clarity and the strength of his results had improved, and Hardy was simply stunned by what he saw. He immediately invited Ramanujan to come visit him in Cambridge, and the two became lifelong friends. 

Alas, his life was very short: Ramanujan died at age 32 of tuberculosis (or possibly of a liver parasite; recent research suggests this may have been his underlying condition), less than six years after his letter to Hardy.

When we talk about people whose early death was a tremendous loss to humanity, there are few people for whom it's as true as Ramanujan, and a recent discovery in his papers has just underlined why.

This discovery ties together two stories separated by centuries: The "1729" story, and the great mystery of Pierre Fermat's last theorem.

The 1729 story comes from a time that Hardy came to visit Ramanujan when he was ill. In Hardy's words: 

"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'"

This has become the famous Ramanujan story (and in fact, 1729 is known to this day as the Hardy-Ramanujan Number), because it's just so ludicrously Ramanujan: he did have the reputation of being the sort of guy to whom you could mention an arbitrary four-digit number, and he would just happen to know (or maybe figure out on the spot) some profound fact about it, because he was just that much of a badass.

The other story is that of Fermat's Last Theorem. Pierre de Fermat was a 17th-century French mathematician, most famous for a theorem he didn't prove. In 1637, he jotted down a note in the margins of a book he had, about a generalization of the Pythagorean Theorem.

From Pythagoras, we know that the legs and hypotenuse of a right triangle are related by a²+b²=c². We also know that there are plenty of sets of integers that satisfy this relationship -- say, 3, 4, and 5. Fermat asked if this was true for higher powers as well: that is, when n>2, are there any integers a, b, and c such that aⁿ+bⁿ=cⁿ? He claimed that the answer was no, and that "he had a truly marvelous proof of this statement which was, unfortunately, too large to fit in this margin."

The consensus of mathematicians ever since is that Pierre de Fermat was full of shit: he had no such proof, and was bluffing.

In fact, this statement -- known as Fermat's Last Theorem, as his notes were only discovered after his death -- wasn't proven until 1995, when Andrew Wiles finally cracked it. Wiles' success was stunning because he didn't use any of the traditional approaches: instead, he took (and significantly extended) a completely unrelated-seeming branch of mathematics, the theory of elliptic curves, and figured out how to solve this. That theory is also at the heart of much of modern cryptography, not to mention several rather unusual bits of physics. (Including my own former field, string theory)


And so these two stories bring us to what just happened. A few months ago, two historians digging through Ramanujan's papers were amused to find the number 1729 on a sheet of paper: not written out as such, but hidden in the very formula which expresses that special property of the number, 9³+10³=12³+1. 

What turned this from a curiosity into a holy-crap moment was when the rest of the page, and the pages that went with it, suddenly made it clear that Ramanujan hadn't come up with 1729 at random: that property was a side effect of him making an attempt at Fermat's Last Theorem.

What Ramanujan was doing was looking at "almost-solutions" of Fermat's equation: equations of the form aⁿ+bⁿ=cⁿ±1. He had developed an entire mechanism of generating triples like these, and was clearly trying to use this to home in on a way to solve the theorem itself. In fact, the method he was using was precisely the method of elliptic curves which Wiles ended up using to successfully crack the theorem most of a century later.

What makes this completely insane is this: Wiles was taking a previously-separate branch of mathematics and applying it to a new problem.

But the theory of elliptic curves wasn't even invented until the 1940's.

Ramanujan was making significant progress towards solving Fermat's Last Theorem, using the mathematical theory which would in fact prove to be the key to solving it, while making up that entire branch of mathematics sort of in passing.

This is why Ramanujan was considered one of the greatest badasses in the history of mathematics. He didn't know about 1729 because his head was full of random facts; he knew about it because, oh yes, he was in the middle of doing yet another thing that might restructure math, but it didn't really solve the big problem he was aiming at so he just forgot about it in his stack of papers.

I shudder to imagine what our world would be like if Ramanujan had lived a longer life. He alone would probably have pushed much of mathematics ahead by 30 or 40 years.


If you want to know more about elliptic curves, Fermat, and how they're related, the linked article tells more, and links to more still. You can also read an outline of Ramanujan's life at https://en.wikipedia.org/wiki/Srinivasa_Ramanujan , and about Fermat's Last Theorem (and why it's so important) at https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem .
Ramanujan's manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat's last theorem appears further up: α3 + β3 = γ3 + (-1)n. Image courtesy Trinity College library. Click here to see a larger ...
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Drones abd gopros have changed videography forever.

https://youtu.be/RuVnAwkuTVw?t=2m36s
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My locality appeared in the papers today.
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Flipkart doesn't understand India. You've been good so far, bye bye.

FYI, I've 3g connection from airtel and tried this from the center of Bangalore city - Richmond circle.
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See the second video first - http://www.dailyo.in/arts/sujoy-ghosh-ahalya-radhika-apte-alma-rodrigo-blaas-pixar-soumitra-chatterjee/story/1/5148.html

Also, There are 1 too many flaws in Sujoy's film for it to be getting this much attention.
The music and the art direction are equally good in and all in all it is neat cinematic venture.
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Syriza can't use Germany's 1953 debt restructuring as a precedent for its debt relief demand.
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Have them in circles
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School of Computer Science Support Staff. Reasonable Person Principle: Everyone will be reasonable. Everyone expects everyone else to be reasonable. No one is special. Do not be offended if someone suggests you are not being reasonable. Reasonable people think about their needs, and the needs of ...
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"Prashant appeared to be a good boy, but it seems that he has no remorse about Akhlaq's death. Instead, he asked us that after the partition, when it had been decided that Hindus will stay here and Muslims will go to Pakistan, why did Gandhi and Nehru ask Muslims stay back in India? I couldn't help but feel dismayed. These are the typical beliefs that keep the pot of communalism boiling.
 
Prashant and I had a heated argument, but I lost. People like us are losing arguments every day. All I could do was ask Prashant to reconsider his views, read a few more books, but he looked self-assured that whatever he knows is true. It is final. I wonder who would have taught Prashant all this? Did someone come amongst these young men well before they coagulated into the mob of that Monday night? Who are those people who have left young men like Prashant to be misled by the purveyors of false histories? Who are those scholars who have left the Prashants of our villages behind to submit their own useless PHDs to earn accolades in foreign universities?
 
We are not understanding what is happening around us. We are not being able to make others understand. The sparks have been spread across our villages. Young men with their half-baked sense of history want me to pose with them for selfies, but are not willing to even consider my appeal that they give up their violent ideals. Our politics has become a collective of opportunists and cowards."
 
I had gone to Dadri to cover Mohammad Akhlaq's death. On the way back, I felt I was carrying another corpse inside me.
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This is the saddest thing I've read in a long time. Read it >2 days ago and been upset since. :-(
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"Goyal was told to major in civil engineering. Other students could learn about databases. For him, hydrology awaited"
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Partition displaced fifteen million people and killed more than a million. Credit Photograph by Margaret Bourke-White / LIFE Picture Collection / Getty
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Always I have a strange sensation when I read something that I had written some time previously...I recognize it ofcourse, but not wholly; it seems almost that I was reading some familiar piece by another, who was near to me and yet who was different...perhaps that is the measure of the change that has taken place in me
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