Ramanujan’s Astonishing Knowledge of 1729

Question: What is the connection between Ramanujan’s number 1729 and Fermat’s Last Theorem? For the answer, read on.

The story of how Srinivasa Ramanujan responded to G. H. Hardy’s comment on the number of a taxi is familiar to all mathematicians. With the recent appearance of the film The Man who Knew Infinity, this curious incident is now more widely known.

Result of a Google image search for “K3 Surface”.

Visiting Ramanujan in hospital, Hardy remarked that the number of the taxi he had taken was 1729, which he thought to be rather dull. Ramanujan replied “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

It is a trivial matter to show that

$\displaystyle 1729 = 9^3 + 10^3 = 12^3 + 1^3 \,.$

But how could Ramanujan possibly have known this abstruse fact? This question has mystified mathematicians for almost a century.

Diophantine Equations

At first glance, it is remarkable that Ramanujan knew the properties of the number 1729. Material recently uncovered in the library of Trinity College, Cambridge shows that the story was not simply a charming tale dreamed up by Hardy. Ramanujan came upon the number 1729 during a search for integer “near-solutions” of the diophantine equation

$\displaystyle x^3 + y^3 = z^3 \,.$

Pierre Fermat had claimed in 1637 an extraordinary proof that the equation

$\displaystyle x^n + y^n = z^n$

has no non-trivial solutions in integers $(x, y, z)$ for any integer ${n>2}$ . This problem has a long and fascinating history, leading to the successful proof by Andrew Wiles published in the May 1995 issue of the Annals of Mathematics.

The general Fermat problem taxed the minds of some of the greatest mathematicians for centuries. The special case of the Fermat equation for ${n=3}$ was proved by Leonhard Euler in 1770. Euler used the method of infinite descent.

In a recent paper (Ono and Trebat-Leder, 2015) on arXiv.org, Prof Ken Ono of Emory University in Atlanta, Georgia, pointed out that Ramanujan had been studying Euler’s diophantine equation

$\displaystyle x^3 + y^3 = z^3 + w^3 \ \ \ \ \ (1)$

and had come across a solution involving the number 1729 in this context.

[Aside: Ramanujan’s work foreshadowed deep structures and phenomena that are of fundamental importance in modern algebraic geometry and number theory. In particular, he discovered a geometric structure called a K3 surface. K3 surfaces are complex smooth minimal complete surfaces that have a role in string theory. For a sample, see the Figure at the head of this post.]

In his Lost Notebook (§8.5 of Andrews and Berndt, 2013), Ramanujan presented a stunning method for generating an infinite family of solutions to (1). His method involved the expansion of three rational functions at zero and at infinity. The integer 1729 emerges from this process.

Page from Ramanujan’s Lost Notebook. Image credit: Trinity College Cambridge. Reproduced from Ono, 2015.]

As is clear from the formulas in the Figure above, Ramanujan defined three rational functions and expanded them about the origin in powers of ${x}$ and about infinity in powers of ${\xi=1/x}$ . The functions and ${x}$ -expansions are

$\displaystyle \begin{array}{rcl} \frac{1 + 53 x + 9 x^2}{1 - 82 x - 82 x^2 + x^3} &=& a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots \\ \frac{2 - 26 x - 12 x^2}{1 - 82 x - 82 x^2 + x^3} &=& b_0 + b_1 x + b_2 x^2 + b_3 x^3 + \cdots \\ \frac{2 + 8 x - 10 x^2}{1 - 82 x - 82 x^2 + x^3} &=& c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots \end{array}$

The expansions in ${\xi=1/x}$ are

$\displaystyle \begin{array}{rcl} \frac{\xi^3 + 53 \xi^2 + 9\xi}{\xi^3 - 82\xi^2 - 82\xi + 1} &=& \alpha_0\xi + \alpha_1\xi^2 + \alpha_2\xi^3 + \alpha_3\xi^4 + \cdots \\ \frac{2\xi^3 - 26\xi^2 - 12\xi}{\xi^3 - 82\xi^2 - 82\xi + 1} &=& \beta_0 \xi + \beta_1 \xi^2 + \beta_2 \xi^3 +\beta_3\xi^4 + \cdots \\ \frac{2\xi^3 + 8\xi^2 - 10\xi}{\xi^3 - 82\xi^2 - 82\xi + 1} &=& \gamma_0\xi + \gamma_1\xi^2 + \gamma_2\xi^3 +\gamma_3\xi^4 + \cdots \end{array}$

(the indices follow Ramanujan’s choice). The first few coefficients ${\{a_n,b_n,c_n\}}$ and ${\{\alpha_n,\beta_n,\gamma_n\}}$ are listed in Table 1.

Ramanujan-1729-Table

Ramanujan concluded that, for each set of coefficients, the following relations hold:

$\displaystyle \begin{array}{rcl} a_n^3 + b_n^3 &=& c_n^3 + (-1)^n \\ \alpha_n^3 + \beta_n^3 &=& \gamma_n^3 + (-1)^n \end{array}$

We see that the values ${\alpha_0 = 9}$ , ${\beta_0 = -12}$ and ${\gamma_0 = -10}$ in the first row correspond to Ramanujan’s number 1729.

The expression of 1729 as two different sums of cubes is shown, in Ramanujan’s own handwriting, at the bottom of the document reproduced above. This is just a single member of an infinite family of solutions.

The page from the Lost Notebook resolves the question of how Ramanujan knew about the number 1729. But it raises a much greater question: how did he come upon the three extraordinary rational functions that are presented in the notebook? That is a question for another day.

* * *

An earlier post on Ramanujan’s Lost Notebook is here.

Sources

${\bullet}$ Andrews, G. E. and B. C. Berndt, 2013: Ramanujan’s Lost Notebook, Part IV, Springer, New York, 2013.

${\bullet}$ Freeman L., 2005: Fermat’s Last Theorem: Proof for n = 3. Blog presenting the story of Fermat’s Last Theorem and Wiles’ proof in an accessible manner.

${\bullet}$ Ono, Ken and Sarah Trebat-Leder, 2015: The 1729 K3 Surface. arXiv:1510.00735v4 [math.NT] 19 Oct 2015.

${\bullet}$ Silverman, Joseph H., 1993: Taxicabs and Sums of Two Cubes. Amer. Math. Mon., 100, 331-340.

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