Maths/Hist: The curious number 1729

The number 1729 is special for a number of reasons. One is that it is usually said to be the first number that is the sum of two cubes in two different ways. Can you find the two distinct solutions to the equation x³ + y³ = 1729? Srinivasa Ramanujan was a self-taught Indian mathematician, and he sent a bundle of his most interesting results to Godfrey Hardy in Cambridge. The work Ramanujan had done in India was so astounding that Hardy arranged for Ramanujan to travel to Cambridge to work with him.

They say two other mathematicians received Ramanujan’s material first, but they sent it back without comment. Hardy decided an unknown mathematician of genius was more likely than a fraud of the genius needed for a clever fraud, and assumed that Ramanujan was genuine.

Ramanujan died at the age of 33 from tuberculosis, after a long illness. On one occasion, Hardy visited his sick colleague, and while making conversation, Hardy mentioned the number of his taxicab, his favourite form of transport. It had, said Hardy, a rather dull number, 1729. Ramanujan disagreed. “Oh no, Hardy, it is a very interesting number. It is the first number that is the sum of two cubes in two different ways!” Ramanujan was referring here to the fact that 1729 is the sum of one cubed and twelve cubed, and also the sum of nine cubed and ten cubed.

The two mathematicians then went on to discuss the fourth powers equivalent, but that has no part here. There is a solution, by the way, with 133 and 134 being the numbers on one side: the rest I leave to you.

1729 is one of a special group of numbers called Carmichael numbers, which are important in number theory, but we will ignore them for now. Hardy was probably trying to find out if Ramanujan had discovered Carmichael numbers in his intuitive way, but he got a surprising answer.

The number 1729 is an unusual Carmichael number, because its three factors (7, 13 and 19) are in arithmetic progression: we have known since the mid-1990s that there is an infinite number of Carmichael numbers, but is there an infinite number of them with factors in AP? Probably…

An oddity that struck me while I was editing this chapter: there is a smaller number that is the sum of two cubes in two different ways. That number is 91, and while 1729 = 7 x 13 x 19, 91 = 7 x 13. We will meet 91 in other places here, but 91 = 4³ + 3³ = 6³ + (-5)³.

Now, about Carmichael numbers, there are 20,138,200 Carmichael numbers between 1 and 10²¹, and the first seven are:

561 (3 x 11 x 17); 1105 (5 x 13 x 17); 1729 (7 x 13 x 19); 2465 (5 x 17 x 29);
2821 (7 x 13 x 31); 6601 (7 x 23 x 41); and 8911 (7 x 19 x 67).

In the early days on the internet, while discussing Carmichael numbers, I noted that two in this sample (1729 and 2465) had factors in arithmetical progression. These were briefly dubbed Macinnis numbers, but they led to nothing interesting, so we forgot about them.

One of my students threw 1105 at me, asking what I thought of it. I replied that it was a Carmichael number, but he thought it was better described as the product of the first three primes of the form 4n+1.

I should have known that he was laying a trap for me. “It’s perhaps more to the point that it is the sum of two squares in three different ways,” I said.

He fired back much too quickly. “Four ways, actually!”

I constructed this spreadsheet, and saw that he was correct:

There is enough here for the clever reader to work out how this goes on, and what I was doing.

Let me say that I love having students like that.

They are also the ones who enjoy challenges like the game of Srinivasa.

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