### Douglas Drumond

Shared publicly -**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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