Ramanujan and Hardy's taxi

G H Hardy, the famous British mathematician, brought Ramanujan, a poor Indian who was a natural mathematical genius, to work with him at Cambridge University. They had a very important and creative mathematical partnership.

The British climate was bad for Ramanujan. He got tuberculosis. As he lay dying in hospital, Hardy went to visit him.

As he entered the room he said, "The number of my taxi was 1729. It seemed to me rather a dull number." To which Ramanujan replied, "No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Ramanujan died at the age of 33.

Is the number in the story right?

Is 1729 the smallest number expressible as the sum of two cubes in two different ways?

It is obvious it is expressible as the sum of two cubes in two different ways, but is it the smallest such number?

You can treat this as a mathematical proof or as a programming exercise.

Obviously we can think of some other questions:

If 1729 is the smallest such number, what is the second smallest? the nth smallest?

Can we write a program to find this?

What do we mean by "different ways"?

What about the most general statement of the problem:

What is the kth smallest number that can be expressed as the sum of the nth powers of m numbers in r different ways? Can we find a formula for it? (I doubt it) Can we write a program to do it? Are the results interesting?

Any Computer Science student who wishes to solve this problem may, with my agreement, substitute it for some of their project work on one of the courses they take from me. 

The programming isn't as easy as it might at first seem. Some of these numbers are going to be rather big!

Chris Stephenson

Read the story of Ramanujan and Hardy in Curiosity Corner